There's an algorithm to tell if such a map exists if $N$ and $M$ are compact 3-manifolds. Given a covering map $\partial N\to \partial M$ of degree $n$, any extension to a covering $N\to M$ must be a covering of the same degree $n$. Construct all covers of $M$ of degree $n$ (this amounts to finding all reps. of $\pi_1 M$ to the symmetric group $S_n$), and determine for which of these covers there is a lift of the maps $\partial N\to \partial M$. This reduces the question to whether given a homeomorphism $\partial N \to \partial M$ extends to a homeomorphism $N\to M$ by replacing $M$ with the $n$-fold covers together with the lifts of the boundary maps (in fact this part of the argument works in any dimensions). So we will now restrict to the case that the map $\partial N\to \partial M$ is a homeomorphism.

I'll now assume that $N$ and $M$ are 3-dimensional with non-trivial boundary (otherwise you're just asking for the homeomorphism problem), and irreducible (one may reduce to the irreducible case in the usual fashion using the Kneser-Milnor decomposition). Then $N$ and $M$ are Haken. Put a boundary pattern in $\partial N$ which has no non-trivial automorphisms, and use the homeomorphism to $\partial M$ to transfer the boundary pattern to $\partial M$.

By Theorem 6.1.6 of Matveev's book, there is an algorithm to tell if there is a homeomorphism of $N$ with $M$ which preserves the boundary pattern. This algorithm will then tell if the homeomorphism $\partial N\to \partial M$ extends to a homeomorphism $N\to M$ since the boundary pattern has no automorphisms.

I don't expect this homemorphism extension problem to be solvable in higher dimensions, unless possibly one restricts to some very special class of manifolds.

There's an algorithm to tell if such a map exists if $N$ and $M$ are compact 3-manifolds. Given a covering map $\partial N\to \partial M$ of degree $n$, any extension to a covering $N\to M$ must be a covering of the same degree $n$. Construct all covers of $M$ of degree $n$ (this amounts to finding all reps. of $\pi_1 M$ to the symmetric group $S_n$), and determine for which of these covers there is a lift of the maps $\partial N\to \partial M$. This reduces the question to whether given a homeomorphism $\partial N \to \partial M$ extends to a homeomorphism $N\to M$ by replacing $M$ with the $n$-fold covers together with the lifts of the boundary maps (in fact this part of the argument works in any dimensions). So we will now restrict to the case that the map $\partial N\to \partial M$ is a homeomorphism.

I'll now assume that $N$ and $M$ are 3-dimensional with non-trivial boundary (otherwise you're just asking for the homeomorphism problem), and irreducible (one may reduce to the irreducible case in the usual fashion using the Kneser-Milnor decomposition). Then $N$ and $M$ are Haken. Put a boundary pattern in $\partial N$ which has no non-trivial automorphisms, and use the homeomorphism to $\partial M$ to transfer the boundary pattern to $\partial M$.

By Theorem 6.1.6 of Matveev's book, there is an algorithm to tell if there is a homeomorphism of $N$ with $M$ which preserves the boundary pattern. This algorithm will then tell if the homeomorphism $\partial N\to \partial M$ extends to a homeomorphism $N\to M$ since the boundary pattern has no automorphisms.

I don't expect this homemorphism extension problem to be solvable in higher dimensions, unless possibly one restricts to some very special class of manifolds.