>>The question has already been answered by Joel Hamkins. Here I want to elaborate on Joel's answer (see Proposition A below) so as to point out a nontrivial variant of the question (see Question A below). 
 
>>          In this edit **Proposition A, and the Answers (1) and (2)** have been modified.

**Proposition A.** *Suppose* $M$ *is a nonstandard model of* $I\Delta_0$.  *If* $M$  *can be end extended to some* $M' \models PA$, *then*:

**(a)** $M \models B\Sigma_1$, *and*

**(b)** $M \models$ Con($I\Sigma_c$) *for some nonstandard* $c\in M$.

**Explanation.** The scheme $B \Sigma_1$ is known as the $\Sigma_1$-collection scheme, it is well-known that if a model $M$ of $I\Delta_0$ can be end extended to a model $M'$ of $I\Delta_0$, then $M$ satisfies $B \Sigma_1$ (see Kaye's textbook on models of PA). It is also well-known that $PA$ proves the consistency of each of its finite subtheories, and that there is a recursive sequence of arithmetical sentences $\langle\sigma_n: n\in\omega \rangle$ such that $PA$ proves "each $\sigma_x$ axiomatizes $I\Sigma_x$" (again, as detailed in Kaye's text). Note that statements of the form Con($ \sigma_n$) are $\Pi_1$-statements and therefore their truth is inherited by initial segments closed under addition and multiplication.  Hence by the overspill principle, (b) holds (Note that the overspill principle is applied within $M'$ to find the desired $c$ (which, without loss of generality, can be located in $M$).

In light of the above, the "right" question to ask is:

**Question A.** *Is the converse of Proposition A true?*

**Answer (1).** The answer to Question A is in the positive, *if the assumption that $M$ is a model of $I\Delta_0$ is strengthened to the assumption that $M$ is a model of* $I\Sigma_1$. This follows from the nontrivial fact that every model of $I\Sigma_1$ (even uncountable ones) has an expansion to a model of $WKL_0$ (a result due originally due to Peter Hájek, and recently revisited in [this paper][1] of Tin Lok Wong). Note that the compactness theorem of first order logic holds in $WKL_0$, so an expansion of $M$ to a model of $WKL_0$ can be used to build an end extension of $M$ that satisfies $PA$.

**Answer (2).** The answer to Question A is also in the positive *if we add the requirement that $M$ is countable, and exponentiation is a total function in* $M$ (by Corollary 2.8 & Proposition 3.4 of [this paper][2] of Wong and myself). 

>>I suspect that (1) and (2) describe the current "state of the art" in relation to Question A.

  


  [1]: http://www.logic.univie.ac.at/~wongt9/papers/ew-wkl0star.pdf
  [2]: http://www.logic.univie.ac.at/~wongt9/papers/ew-wkl0star.pdf/www.logic.univie.ac.at/