I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absolute neighbourhood retracts, and thus have the homotopy type of CW-complexes. This is briefly stated in corollary 1 of Milnor's article *On spaces having the homotopy type of a CW-complex* (published in Trans. Amer. Math. Soc. 90, 1959, pages 272-280). An extensive discussion of these matters is given in the thesis *A topological manifold is homotopy equivalent to some CW-complex* by Aasa Feragen. For the specific case of compact manifolds, an elementary proof is given in the appendix of Hatcher's book *Algebraic topology* (see corollary A.12 there).

We can now prove the result you state. Let $Y$ be a CW-complex admitting a homotopy equivalence $f : Y \to \partial M$ to the topological manifold $\partial M$. Denote by $j : \partial M \to M$ the inclusion of the boundary of $M$, and factor the composition
$$ Y \overset{f}{\longrightarrow} \partial M \overset{j}{\longrightarrow} M $$
as the inclusion of a sub-CW-complex $i : Y \to X$ followed by a weak equivalence $h : X \to M$. The weak equivalence $h$ is a homotopy equivalence since both its domain and its target have the homotopy type of CW-complexes.

Finally, let me briefly justify why we can take $i : Y \to X$ to be the inclusion of a subcomplex of a CW-complex. The usual construction of the factorization above produces $(X,Y)$ as a *relative* CW-complex, i.e. $X$ is obtained from $Y$ by adding cells in increasing order of their dimension. Now observe that this construction can be modified to inductively build $X$ as a CW-complex itself: at each stage of the construction of $X$, we can replace the attaching maps of the cells with homotopic cellular maps using the cellular approximation theorem.