The nonreciprocal photon transmission between two optical modes which are nonlinearly coupled in a cross-Kerr type is investigated by using a damped auxiliary optical mode as their common engineered reservoir. The optical nonreciprocity is created by quantum interference between the nonlinear coupling induced by the cross-Kerr nonlinearity and the dissipative coupling mediated by the adiabatic elimination of the auxiliary optical mode. The photon transmission probabilities in forward and backward directions for the relative phase between the two couplings are symmetric about the resonance with the probabilities for the phase , and are exchange symmetric with those in the case of the phase . By tuning the dissipative coupling, in addition to the photon nonreciprocity being improved, the nonreciprocal photon amplification is produced. Meanwhile, it is found that not only can the cross-Kerr nonlinearity lead to a broadband optical nonreciprocity, but also it effectively produces the optical squeezing in the present scheme. The combination of the photon nonreciprocity with the optical squeezing will be useful in the realization of the quantum information processing with noise reduction.