# Does the normal bundle of a manifold depend on embedding?

In the proof of unoriented cobordism ring being isomorphic to homotopy group of Thom spectra, one considers a large enough dimensional Euclidean space where a given manifold has all the embeddings isotopic. This is needed to show the Pontrjagin Thom collapse map does not depend on the embedding.

Here is my question: Why does one need isotopy of embeddings? Given a Euclidean space any embedding will have normal bundle isomorphic to a trivial bundle modulo the tangent bundle of the given manifold. It does not depend on embedding in that particular Euclidean space. Any help will be appreciated.