Background: When proving that the group of $k$-isogenies $\mathrm{Hom}_k(A,B)$ between two abelian varieties is finitely generated, one first shows that the Tate map $$\mathbb{Z}_\ell\otimes_{\mathbb{Z}} M \to \mathrm{Hom}_{\mathbb{Z}_\ell}(T_\ell A,T_\ell B)$$ is injective. Since each Tate module is free of finite rank over $\mathbb{Z}_\ell$, it follows that the localization $M_\ell$ is $\mathbb{Z}_\ell$-finite. One then uses a little trick to deduce the $\mathbb{Z}$-finiteness of $M$ itself. (See Silverman I, for example.)

The above proof needs only a single prime $\ell$, but disregarding issues of the characteristic of the field (which are apparently surmountable) we actually have an injective Tate map at every prime. Thus...

Question: Can the $\mathbb{Z}$-finiteness of $M$ be deduced directly from the $\mathbb{Z}_\ell$-finiteness of $M_\ell$ for all primes $\ell$?

One can consider this a question about general torsion-free abelian groups $M$. A non-counterexample to keep in mind is $M=\mathbb{Z}[1/p]$, for which $M_\ell$ is $\mathbb{Z}_\ell$-finite for all $\ell\neq p$.

(A google search shows that there is actually quite a body of literature on torsion-free abelian groups, so perhaps the answer to this question is well-known, but I'm not sure where to look...)