The determinantal ideal It is the ideal generated by all the t-minors of a generic matrix. With the minimal generators of the initial ideal of the product \(I_{t_1}\cdots I_{t_r}\) not explicitly known and its Gröbner basis computationally intractable, in this paper we study the degree bounds on the minimal generators of the initial ideal of \(I_{t_1}\cdots I_{t_r}\) and thus on the polynomials in its reduced Gröbner basis. Specifically, we show that an upper bound on the degrees of the minimal generators is \(t_1+\sum _{k=2}^{r}k(t_k-1)\) by analyzing the divisibility between monomials in the initial ideal via special functions applied to increasing decompositions of the monomials. For the special case of products IaIb of two determinantal ideals, we prove the existence of minimal generators of each degree between a + b and the presented upper bound and thus fully characterize the degrees of the polynomials in the reduced Gröbner bases of such products.