A variation of your question has a positive answer. If you take any compact manifold that is a smooth bundle over another compact manifold $\pi : M \to N$, there is a smooth embedding

$$f : M \to N \times \mathbb R^k$$

giving a commutative diagram i.e. such that $pr_1 \circ f = \pi$, for some $k$ usually fairly large. 

There's a lot of ways of proving this, but if you know about the theory of classifying spaces it follows from the observation that if $F$ is the fibre of $f$, then $BDiff(F)$$ can be thought of as the space

$$Emb(F, \mathbb R^\infty) / Diff(F)$$

The space is the limit $Emb(F, \mathbb R^\infty) = \lim_k Emb(F, \mathbb R^k)$ induced by the inclusions $\mathbb R^k \to \mathbb R^{k+1}$. 

The proof is an old argument of Whitney's, basically a variation of the weak Whitney embedding theorem, that $Emb(F, \mathbb R^k)$ is highly connected provided $k$ is large.   Off the top of my head I don't remember the connectivity, but it's about $\lceil \frac{k}{2} -dim(F) - 1\rceil$, i.e. the greatest integer less than or equal to the above.   Making $k$ small is analogous to the question of asking the question of what is the smallest dimensional Euclidean space a manifold embeds into.  You are basically asking that question "with parameters". 

In your specific case the vector space is one dimension larger than your fiber sphere $S^n$.   So from this point of view your question is asking about the pseudoisotopy diffeomorphism group of the sphere.  When $M$ is a circle, such embeddings do not exist due to Cerf's pseudoisotopy theorem.  You are interested in if the map from the pseudoisotopy diffeomorphism group to the vanilla diffeomorphism group can be non-trivial $PDiff(S^n) \to Diff(S^n)$.  Some people would also phrase the question of if you can construct (families) of diffeomorphisms of $S^n$ that are pseudoisotopically trivial.    I believe the diffeomorphisms of $S^4$ constructed by Watanabe  are pseudoisotopically trivial.    In his original paper he has a non-constructive argument, but he may very well have a constructive argument in follow-on papers. 

https://www.math.kyoto-u.ac.jp/~tadayuki.watanabe/Diff_S4.pdf

I have a "soon to appear" paper where we write down the null pseudo-isotopies constructively, i.e. you can see them at a point-set level.  

So Watanabe's construction would answer your question when $N = S^2$ and the fiber is $S^4$.   But there are several papers published in the past few years that likely have analogous results for $N$ a broad range of spheres and the fiber being high-dimensional spheres.  I would start maybe looking at the papers of Alexander Kupers.