This question was studied in a paper by Browder who proves the following.
Thereom. Consider any Serre fibration of $F\to S^n\to B$ where $F$, $B$ are connected polyhedra. Then $F$ is homotopy equivalent to $S^1$, $S^3$, or $S^7$. If $F=S^1$, then $B$ is homotopy equivalent to $CP^k$ with $2k+1=n$. If $F=S^7$, then $B$ is homotopy equivalent to $S^8$.
One expects that if $F= S^3$, then $B$ is homotopic to a quaternionic projective space of a suitable dimension but Browder mentions (if I am reading is correctly) that this is not true and there are other examples.