I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection.

Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ equipped with some connection $D$, I want to find an orthogonal bundle $W$ with a *flat* connection $D'$ such that $V\subset W$ is a subbundle and $D$ is induced from $D'$.
Here by "induced" I mean the following: given a section $s$ of $V$ and a vector field, the covariant derivative of $s$ using $D$ should be the composition of the covariant derivative of $s$ (seen as section of $W$) using $D'$ with the projection onto $TV$ using the orthogonal structure of $W$.

A special case is when $M$ is Riemannian. For $V=TM$ and $D$ the Levi-Civita connection, the answer to the question is Nash's embedding theorem: there is an embedding of $M$ into some $\mathbb{R}^N$ such that the connection $D$ is induced by the trivial connection $d$ on $\mathbb{R}^N$.

Is there a result of this kind? Any references?