transverse homotopy

Question

Let $f,g:M \rightarrow N$ two smooth maps between smooth manifolds that are smoothly homotopic by $F$. Suppose also that $f$ and $g$ are transverse to a submanifold $A$ of $N$. I know that transverse maps are dense in the weak and strong topologies for the space of maps $C^{\infty}(M \times I, N)$ so its always possible to approximate the map $F$ by a map transverse to $A$. I also wish to find a transverse approximation to $F$ that is still a homotopy between $f$ and $g$, anyone knows if this is possible?