No. Let $M$ be $S^1\times S^2$ and let $K$ be homeomorphic to $S^1$ and chosen such that in the universal cover $\tilde M=\mathbb R\times S^2$ there are two liftings $K_1$ and $K_2$ of $K$ which are linked together. If $U$ existed then these would be contained in two disjoint liftings $U_1$ and $U_2$, therefore unlinked.

No. Let $M$ be $S^1\times S^2$ and let $K$ be homeomorphic to $S^1$ and chosen such that in the universal cover $\tilde M=\mathbb R\times S^2$ there are two liftings $K_1$ and $K_2$ of $K$ which are linked together. If $U$ existed then these would be contained in two disjoint liftings $U_1$ and $U_2$ of $U$, therefore unlinked.