We study the Sarason's problem for corresponding Hankel product problem on the Fock space. We completely characterize the boundedness of the Hankel product when one symbol is a finite linear combination of reproducing kernel functions and the other is a polynomial. Moreover, a necessary and sufficient condition to the boundedness of the Hankel product is proposed for the case that one symbol is the product of a linear combination of reproducing kernels and a polynomial. We also use a different method to reprove the semi-commutating Toeplitz operators on the Fock space.