Let $M, N, X$ are compact manifolds. Let $f_1:M \rightarrow X$ and $g_1: N \rightarrow X$ be any two embeddings. Is it always possible to find embeddings $f_2 $ homotopic to $f_1$ and $g_2$ homotopic to $g_1$ such that images of $f_2$ and $g_2$ are disjoint? I think not, but can't find a counter example.