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" which in this case requires $4n \le m$ (approximately).${}^\dagger$

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

${}^\dagger$ **Added Later:** According to Steimle, the "stable range" for the fibering problem is more complicated than what I wrote above. Rather than write it down, let me refer to his paper for the actual range.