The existence of a simply-connected neighborhood of a contractible loop

Question

Let $M$ be a smooth manifold with a point $x_0$ on $M$ and a smooth loop $\gamma$ at $x_0$. If $\gamma=0$ in $\pi_1(M,x_0)$, then can we find a simply-connected open set $U$ around $x_0$ such that $\gamma \subset U$?

Answer 1

There exist surjective maps $S^1 \to S^1$ of degree zero, for example the concatenation (as loops) of the identity and its reverse. They are zero in the fundamental group, but the only open set that contains them is $S^1$ which isn't simply connected.