For a unitary matrix X of order n over the field of complex numbers and an entire function φ belonging to the Fock space F2: = F2(Cn) , we define an integral operator on F2(Cn) of the form (HφXf)(z)=∫Cnf(w)φ(z+X∗Xw¯)ezw¯dλ(w).Here dλ(z)=π-ne-|z|2dz is a Gaussian measure on Cn. We characterize all the symbols φ for which the operator HφX is bounded. Next, we consider integral operator on F2 defined by (Rφf)(z)=∫Cnf(w)φ(z⋆w¯)dλ(w)for φ∈ F2, where ⋆ is a coordinatewise multiplication. We give a complete characterization for the symbols φ∈ F2(Cn) so that the operator Rφ is bounded on F2. In addition to boundedness, we also obtain some fundamental results for the operators HφX and Rφ such as normality, the C∗-algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections BX={HφX∈B(F2):φ∈F2},R={Rφ∈B(F2):φ∈F2},respectively. © 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.