Finding a manifold that covers a compact and respect some properties

Question

Let $M^n$ ($n=3$ if needed) be a smooth manifold and $K \subset M$ be a compact submamifold diffeomorphic to the $n$-disk, satisfying $K \subset U$, for some chart $(\varphi,U).$

Consider a compact submanifold $N^n\subset M^n $ with boundary such that

1. $\text {Int }N^n \cap \text {Int}K \neq \emptyset,$
2. $N^n \cap K$ is connected.

I would like to know if there exists a compact smooth manifold $L\subset U $ with boundary such that the following properties hold

1. $K\subset L$,
2. $L \cup N $ is a smooth manifold with boundary,
3. $L\cap N $ is a smooth manifold with boundary.

This result seems true, however, I was not able to prove it. Can anyone help me?

I think it is always possible to do something like the picture below