There is an old result due to Patricia Tulley which claims that this is possible.
P. Tulley, A strong homotopy equivalence and extensions for Hurewicz fibrations, Duke Math. J. 36(3): 609-619 (September 1969).
The main result of the paper is that, with some basic standing assumptions on the spaces involved, two Hurewicz fibrations over the same base are fiber-homotopy equivalent if and only if they are concordant (this being the relation of strong fiber-homotopy equivalence).
This result is then applied to the fibrations $X\rightarrow\ast\leftarrow Y$ when $X,Y$ are homotopy equivalent spaces. The result is a Hurewicz fibring $$p:Z\rightarrow I$$ with $p^{-1}(0)\cong X$ and $p^{-1}(1)\cong Y$.
Although the results for general fibrations require some compactness assumptions, the statement regarding $X,Y$ above is given with no additional assumptions. However, I haven't gone through the paper thoroughly to check all the details.