Tubular neighborhood of $X^k$ compact submanifold with normal bundle $\perp X$ trivial

Question

For $X^k\subset M^n$ compact submanifold with $\perp X$ trivial and set $S^k$ the $k$-sphere. Then there is a function $f:M^n\rightarrow S^k$ such that $X$ is the preimage for a regular value. My question is: the converse is true?

Answer 1

Yes since you can pull back normal bundles and regular values tend to have trivial normal bundles.