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Assume that $M$ is an open connected manifold?

Is there a (not necessarily one point) compactification $\tilde{M}$ of $M$ such that $\tilde{M}$ is a manifold? If yes, can we find such $\tilde{M}$ with zero Euler characteristic?

Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:

By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ which contains $M$ as an open dense subset.

Assume that $M$ is an open connected manifold which admits a manifold compactification. Does $M$ necessarily admit a manifold compactification with zero Euler characteristic?