Yes, one can construct a model of $\mathbb{Z}$ which contains $\mathbb{N}$.
The following construction does not use equivalence classes or embedding technic to make $\mathbb{N}$ subset $\mathbb{Z}$. It rather extends a particular model of natural numbers.
Introduction. Let $(\mathbb{N},\mathrel{+} ,\cdot , \leq)$ be the system of Von Neumann natural numbers.
A natural number $m$ is a particular set with $m$ elements $m = \{0,...,m-1\}$ for $m>0$ and $0=\{\}$.
Subtraction and division can be defined for some pairs of natural numbers.
For $m,n \in \mathbb{N}, m - n$ is the natural number $d$, if there is any, such that $m = n + d.$
For $m,n \in \mathbb{N}, n>0, m\div n$ is the natural number $q$, if there is any, such that $m=n⋅q$.
In what follows I will use the fact that $n\subset\mathbb{N}$ and carry set-theoretical operations on natural numbers.
Construction of Integers
Definition 1. Let $n\in \mathbb{N}$ be a natural number. An opposite number $\overline{n}$ is a subset of $\mathbb{N}$ defined as:
$$\overline{n}:=\begin{cases}0&\text{if } n=0\\ \mathbb{N}\setminus n &\text{if } n\neq 0.\end{cases}$$
The set of all opposite numbers is denoted by $\overline{\mathbb{N}}=\{\overline{n}|n\in\mathbb{N}\}$.
Intuition 1.
An opposite number $\overline{n}$ is a particular set with $n$ elements being missing. Intuitively if we are missing $n$ elements and we receive $n$ then we do not miss anything and therefore we have nothing. This justifies our definition of $\overline{0}=0$.
Definition 2.
We define define the set $\mathbb{Z}$ of integers as
$ \mathbb{Z}=\mathbb{N}\cup \overline{\mathbb{N}}$.
We extend the domain of our definition of $\overline{a}:=\mathbb{N}\setminus a$ to all $a\in\overline{\mathbb{N}}\setminus\{0\}$.
Definition 3.
We define projection functions
$$\mathsf{proj}_0: \mathbb{Z} \to \mathbb{N},a \mapsto a_0:=
\begin{cases}
a, & \text{if } a \in\mathbb{N} \\[2ex]
0, & \text{if } a \in\overline{\mathbb{N}}
\end{cases}
$$
$$\mathsf{proj}_1: \mathbb{Z} \to \mathbb{N},a \mapsto a_1:=
\begin{cases}
\overline{a}, & \text{if } \overline{a} \in\mathbb{N} \\[2ex]
0, & \text{if } \overline{a} \in\overline{\mathbb{N}}
\end{cases}
$$
Definition 4. We define the balance function as follows
$$
\mathsf{bal}: \mathbb{N}\times\overline{\mathbb{N}}\to\mathbb{Z}, (m,\overline{n})\mapsto (m-\mathrm{min}\{m,n\})\cup(\overline{n-\mathrm{min}\{m,n\}}).
$$
The balance function is well-defined as either $m-\mathrm{min}\{m,n\}=0$ or $\overline{n-\mathrm{min}\{m,n\}}=0$.
Intuition 2.
For a natural number $m$ and an opposite number $\overline{n}$ we find a balance between $m$ and $\overline{n}$. If we are missing $n$ elements and we receive $m$ then we have $m-n$ elements if $m<n$, we don’t have or don’t miss any elements if $m=n$ and finally we miss $n-m$ elements if $m>n$.
Definition 5.
We define,$+_\mathbb{Z}, \cdot_\mathbb{Z}$binary operations and $\leq_\mathbb{Z}$ an order on $\mathbb{Z}$ as follows:
$$+_\mathbb{Z} :\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}, (a,b)\mapsto \mathsf{bal}(a_0+b_0,\overline{a_1+b_1})$$
$$\cdot_\mathbb{Z}:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Z}, (a,b)↦(a_0\cdot b_0+ a_1\cdot b_1 )\cup(\overline{ a_0\cdot b_1+ a_1\cdot b_0}) $$
$$ a \leq_\mathbb{Z} b :\Longleftrightarrow a_0 + b_1\leq a_1 + b_0.$$
Again as either $a_0\cdot b_0 + a_1\cdot b_1=0$ or $\overline{a_0\cdot b_1+ a_1\cdot b_0}=0$ and the binary operation $\cdot_\mathbb{Z}$ is well-defined.
Proposition 3.
The binary operations and the order on $\mathbb{Z}$ restricted to natural numbers are the same as the binary operations and the order on $\mathbb{N}$.
As a bonus I present a construction of the rational numbers in the same spirit.
Construction of Rationals
Definition 6. Let $m, n\in \mathbb{N}$ and $n>0$. A ratio of $m : n$ is a subset of $\mathbb{N}$ defined as follows:
$$m:n=(m+n)\div\mathrm{gcd}\{m,n\}\setminus\{m \div\mathrm{gcd}\{m,n\}\}.$$
The set of ratios is the set $\mathbb{L}:=\{m:n|m, n \in \mathbb{N} \text{ and } n\neq 0\}.$ (Ancient and Modern λόγος (lógos) ‘ratio’.)
Intuition 3. We represent natural numbers $m,n$ as intervals $[0; m), [0; n)$ then the ratio of $[0; m) : [0; n)$ is the same as ratio $[0; m), [m; m+n)$. A ratio is a partion of $[0,m+n)$ which we represent by removing the point $m$ from $[0; m+n)$, i.e. $[0; m+n)\setminus \{m\}=[0; m)\cup(m; m+n)$.
Proposition 4.
For coprime natural numbers $m, n \in \mathbb{N}$ and $n>0$
$$m : n := m\cup ((m+n)\setminus (m+1)).$$
For all natural numbers $m \in \mathbb{N}$: $$m : 1 = m.$$
We can now define addition, multiplication and the order on $\mathbb{L}$.
Definition 7.
We define
$$+_\mathbb{L} :\mathbb{L} \times \mathbb{L} \to \mathbb{L}, (a,a')\mapsto (m\cdot n'+m'\cdot n ) : (n\cdot n' )$$
$$\cdot_\mathbb{L} :\mathbb{L}\times \mathbb{L} \to \mathbb{L}, (a,a')\mapsto (m\cdot m'):(n\cdot n' ) $$
$$ a \leq_\mathbb{L} a' :\Longleftrightarrow m\cdot n'\leq m'\cdot n .$$
Definition 8.
Let $a=m:n, a'=m':n'$ be ratios with $a\leq_\mathbb{L}a'$. We define subtraction
$$a'-_\mathbb{L}a:=(m'\cdot n-m\cdot n' ) : (n'\cdot n ). $$
The order condition on the ratios is what is needed for the subtraction of natural numbers to be well-defined.
Proposition 5.
The set of natural numbers is subset of $\mathbb{L}$ and operations of addition, multiplication, subtractions and the linear order on $\mathbb{L}$ extend those on $\mathbb{N}$.
Definition 9. Let $a\in \mathbb{L}$ be a ratio. An opposite ratio $\overline{a}$ is a subset of $\mathbb{N}$ defined as:
$$\overline{a}:=\begin{cases}0&\text{if } a=0\\ \mathbb{N}\setminus n &\text{if } a\neq 0.\end{cases}$$
The set of all opposite ratios is denoted by $\overline{\mathbb{L}}=\{\overline{a}|a\in\mathbb{L}\}$.
Definition 10.
We define the set $\mathbb{Q}$ of rational numbers as
$$ \mathbb{Q}=\mathbb{L}\cup \overline{\mathbb{L}}.$$
We repeat Defintions 3, 4, 5 subsituting $\mathbb{N}$ with $\mathbb{L}$, $\mathbb{Z}$ with $\mathbb{Q}$, and use operations defined on the set of ratios, rather than on the set of natural numbers.
Proposition 6.
$$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}.$$
The binary operations and the order on $\mathbb{Q}$ restricted to natural numbers are the same as the binary operations and the order on $\mathbb{N}$.
The binary operations and the order on $\mathbb{Q}$ restricted to integers are the same as the binary operations and the order on $\mathbb{Z}$.
Definition 11.
We define the fraction function as follows
$$\mathsf{frac}:\mathbb{Z}\times \mathbb{Z} \to \mathbb{Q},$$
$$ (a,b)↦\frac{a}{b}:=(a_0\cdot b_0+ a_1\cdot b_1 ):(b_0^2+b_1^2)\cup\overline{(a_0\cdot b_1+ a_1\cdot b_0):(b_0^2+b_1^2)} $$
Again as either $a_0\cdot b_0 + a_1\cdot b_1=0$ or $\overline{a_0\cdot b_1+ a_1\cdot b_0}=0$ so the fraction function is well-defined.
I don't know how this construction can be extended any further but there are various unique representations of reals as continuous fraction so I think it's a possibility that we can construct real numbers as subsets of $\mathbb{N}$.
As a bonus 2 I recap below Dedekind constructions on $\mathbb{L}$. The following variant is due to Holmes, page 95. Elementary Set Theory with a Universal Set http://math.boisestate.edu/~holmes/holmes/head.pdf.
Construction of Reals
Definition 11.
A magnitude $x$ is a proper initial segment of $\mathbb{L}$ with no greatest element.
The set of magnitudes is the set $$\mathbb{M}:=\{x\subset\mathbb{L}| x \neq \mathbb{L}, \text{for all } a\in \mathbb{L}, a\in x \Longleftrightarrow \text{ for some } b\in x: a<b \}.$$
We define addition, multiplication and the order on $\mathbb{M}$.
Definition 12.
We define
$$+_\mathbb{M} :\mathbb{M} \times \mathbb{M} \to \mathbb{M}, (x,y)\mapsto \{a+ b|a\in x, b\in y\}$$
$$\cdot_\mathbb{M} :\mathbb{M}\times \mathbb{M} \to \mathbb{M}, (x,y)\mapsto \{a\cdot b|a\in x, b\in y\} $$
$$ x \leq_\mathbb{M} y :\Leftrightarrow x\subset y.$$
Definition 13.
Let $x, y$ be magnitudes with $x\leq_\mathbb{L}y$. We define substruction
$$y-_\mathbb{M}x:=\{b-a|b\in y \text{ and } a\notin x\}. $$
Definition 14. Let $x\in \mathbb{M}$ be a magnitude. An opposite magnitude $\overline{x}$ is a subset of $\mathbb{L}$ defined as:
$$\overline{x}:=\begin{cases}0&\text{if } x=0\\ \mathbb{L}\setminus x &\text{if } x\neq 0.\end{cases}$$
The set of all opposite magnitudes is denoted by $\overline{\mathbb{M}}=\{\overline{x}|x\in\mathbb{M}\}$.
Definition 15.
We define the set $\mathbb{R}$ of real numbers as
$$ \mathbb{R}=\mathbb{M}\cup \overline{\mathbb{M}}.$$
From this moment we carry forward the same as for $\mathbb{Z}$ and $\mathbb{Q}$ and again we re-use defintions 3, 4, 5 with analogous changes.
Question.
We can construct a model of $\mathbb{Z}$, $\mathbb{Q}$ where all integers, rationals are subsets of $\mathbb{N}$.
Can we construct a model of $\mathbb{R}$ where all reals are subsets $\mathbb{N}$?
Update 1
I've looked at the constructions of real numbers via continued fractions and I think the answer is one can code real numbers as subsets.
G. J. Rieger. A new approach to the real numbers (motivated by continued fractions). AOh.
Brauceig. Wis. Ge, 33:205–217, 1982.
A. Knopfmacher and J. Knopfmacher. Two constructions of the real numbers via alternating
series. International Journal of Mathematics and Mathematical Sciences, 12(3):603–613,
1989.
Definition 16
Let $a:N\to\mathbb{N}$ be a sequence of natural numbers where $N\in \mathbb{N}$ or $N= \mathbb{N}$ such that $a_{N-1}>1, N\in\mathbb{N}$.
We define recurslively a sequence $ q:N \to\mathbb{N}$, $q_0=1$, $q_1=a_1$, $q_n=a_n\cdot q_{n-1}+q_{n-2}$ for $ n\geq2$. A continued ratio is a subset of $\mathbb{N}$ defined as follows
$$ \lambda(a)=a_0\cup \bigcup_{n\in N\setminus 1} \{a_0+q_n\}$$
Intuition 4
For a continued fraction we have:
$$
a_0+\underset{n\in N\setminus 1 }{\LARGE\mathbb{K}}\frac{1}{a_n} = a_0 +\sum_{n\in N\setminus 1}\frac{(-1)^n}{q_n\cdot q_{n-1}}
$$
and the set $\lambda(a)$ captures all details of the sequence $a$.
The challenge is to explicitly define the arithmetic of continued fraction (analogous to Definition 13, 14, 15) and then re-use Definitions 3, 4, 5 to complete the construction.
http://mathworld.wolfram.com/RegularContinuedFraction.html
Update 2
I've added the extension of the overline operation for completeness sake. A more detailed version of this note is available here.
