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.

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.

Edit: As pointed out by Cihan in the comments, Tulley would later go on to prove the result cited above by showing that any two Hurewicz fibrations over the same base space are fibre-homotopy equivalent if and only if they are strongly fibre-homotopy equivalent.

Theorem (Tulley): Suppose that $B$ is a space and $p_0:E_0\rightarrow B$ and $p_1:E_1\rightarrow B$ are Hurewicz fibrations. Then the following statements are equivalent.

1. There is a map $f:E_0\rightarrow E_1$ which is a homotopy equivalence of spaces and satisfies $p_1\circ f=p_0$.
2. There is a fibre-homotopy equivalence $E_0\simeq_BE_1$.
3. There is a space $Z$ and a Hurewicz fibration $\pi:Z\rightarrow B\times I$ such that $E_0\cong Z|_{B\times 0}$ and $E_1\cong Z|_{B\times1}$. $\quad\blacksquare$

The equivalence of the first and third bullet points above is a well-known result not due to Tulley. The third bullet point is the definition of a strong fibre-homotopy equivalence. In any case, when $B$ is a point, it clearly leads to the desired outcome. For its proof consult:

P. Tulley McAuley, A note on paired fibrations, Proc. Amer. Math. Soc. 34 (1972), 534-540.