No, I don't believe there's a simple solution. But here's an approach to the problem which indicates how it can be fractured up.
Assume $M,N$ are closed and connected. If $f\: M \to N$ is homotopic to a smooth fiber bundle with $M$ and $N$ compact, then the fibers are homotopy finite (i.e., they are homotopy equivalent to a finite complex).
Conversely, it is a result first stated by Quinn (later proved by Gottlieb, and then differently by me) that if $f\: M^m \to N^n$ is such that its homotopy fiber $F$ (at some basepoint in $N$) is homotopy finite, then $F$ is a Poincare duality space of dimension $m-n$. Thus, $f$ gives rise to a fibered surgery problem.
One can approach this problem in two steps:
Step 1: find a block bundle $E \to N$ and a fiber homotopy equivalence $E\simeq M$. This step can be attacked classical surgery techniques (here the dimension of the fiber should be $\ge 6$). What one studies here is the map $\tilde S_N(M) \to \tilde S(M)$ from the fiberwise block structure space to the block structure space.
Step 2: Study the map $S_N(M) \to \tilde S_N(M)$ from the fiberwise structure space to the fiberwise block structure space. This step involves higher algebraic $K$-theory a la Waldhausen. This step is only really understood in the ``concordance stable range'' for $F$, which is about $(m-n)/3$.
The above is only meant to be an outline. I first learned about these ideas from the papers of Weiss and Williams, most notably:
Automorphisms of manifolds. Surveys on surgery theory, Vol. 2, 165–220, Ann. of Math. Stud., 149, Princeton Univ. Press, Princeton, NJ, 2001
An alternative approach which packages Step 1 and Step 2 into a single step is in the third WW paper which can be obtained from Michael Weiss' website.
More recently, see the papers of Wolfgang Steimle, especially
Obstructions to stably fibering manifolds. Geom. Topol. 16 (2012), no. 3, 1691–1724