I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.

Let 
$$
T^n \to M \to T^m 
$$
be a principal torus bundle over a torus. Then $ M $ is a solvmanifold, even a nilmanifold (in fact $ M $ is the total space of a principal torus bundle over a torus if and only if it is a compact nilmanifold for a 2 step nilpotent Lie group, this is theorem 3 of [Palais, Stewart, TORUS BUNDLES OVER A TORUS]). 

What if the bundle is not necessarily principal? Is every torus bundle over a torus a solvmanifold? In other words, if we have a fiber bundle
$$
 T^n \to M \to T^m 
$$
then can we conclude that the total space $ M $ is a solvmanifold? 

This is true for $ n=m=1 $. The compact 2d solvmanifolds are exactly $ T^2  $ and the Klein bottle $ K $. These two are the only circle bundles over the circle. 

For dimension 3 I'm a bit less sure but of course all the principal bundles
$$
 T^1 \to M \to T^2 
$$
are solvmanifolds. And I know that every orientable $ M $
$$
 T^2 \to M \to T^1 
$$
is a solvmanifold. Not sure about mapping torus of orientation reversing maps of $ T^2 $.