Summary: For a unitary matrix \(X\) of order \(n\) over the field of complex numbers and an entire function \(\varphi\) belonging to the Fock space \(\mathfrak{F}^2:=\mathfrak{F}^2(\mathbb{C}^n)\), we define an integral operator on \(\mathfrak{F}^2(\mathbb{C}^n)\) of the form \[ (H_\varphi^X f)(z) = \int_{\mathbb{C}^n} f(w)\varphi (z+X^\ast\overline{Xw})e^{z\overline{w}} d\lambda (w). \] Here \(d\lambda (z) = \pi^{-n} e^{-\vert z\vert^2}dz\) is a Gaussian measure on \(\mathbb{C}^n\). We characterize all the symbols \(\varphi\) for which the operator \(H_\varphi^X\) is bounded. Next, we consider integral operator on \(\mathfrak{F}^2\) defined by \[ (R_\varphi f)(z) = \int_{\mathbb{C}^n} f(w) \varphi (z\star \bar{w})d\lambda (w) \] for \(\varphi \in \mathfrak{F}^2\), where \(\star\) is a coordinatewise multiplication. We give a complete characterization for the symbols \(\varphi \in \mathfrak{F}^2(\mathbb{C}^n)\) so that the operator \(R_\varphi\) is bounded on \(\mathfrak{F}^2\). In addition to boundedness, we also obtain some fundamental results for the operators \(H_\varphi^X\) and \(R_\varphi\) such as normality, the \(C^\ast\)-algebra properties, the spectrum and the compactness. Moreover, we characterize the common reducing subspaces for each of the collections \[ \mathfrak{B}^X = \Big \{H_\varphi^X \in \mathcal{B}(\mathfrak{F}^2) : \varphi \in \mathfrak{F}^2 \Big \}, \mathfrak{R} = \Big \{R_\varphi \in \mathcal{B}(\mathfrak{F}^2) : \varphi \in \mathfrak{F}^2\Big \}, \] respectively.