Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not **homotopic** to a non-surjective map?

**Added**: The motivation is: There is a "mild version" of the Nearby Langrangian conjecture stating: any exact Lagrangian manifold $X \to T^*Y$ has non-zero degree when composed with the projection $T^*Y \to Y$. It is known that the map is always surjective. I am looking at a **possible** inbetween stating that the map cannot be homotoped to a non-surjective map.

not homotopicto a non-surjective map. $\endgroup$4more comments