@Riccardo Not really, but I'm skeptical that any such statement can be true. Two manifolds V→M and V'→M give the same homology classes if their images in $H_*(X;\mathbb{F}_2)$ are equal, but are cobordant iff their images in $MO_*(X)$ are equal, and the map $MO_*(X)→H_*(X;\mathbb{F}_2)$ is extremely far from being injective, even in low codimension. I suspect in your case some geometric considerations may help you, but homotopical methods seem unlikely to give you much
