In this paper, we investigate nonrelativistic, kinetic, linear phase of the filamentation instability when an external magnetic field is present in the direction of the counterstreaming electron beams using Vlasov simulations in 1D-3V space. We first investigate the growth rate of instability. In the linear growth regime, our results correspond to the previous conclusions that with the increase in strength of the ambient magnetic field, there is a suppression of instability. Interestingly, we established that at a critical / threshold magnetic field, Vlasov simulations and particle-in-cell (PIC) simulations differ in their instability behaviour. At this particular magnetic field, there is a complete suppression of the growth of instability in Vlasov results compared to PIC simulations, where a strong growth of instability is shown. It is believed that thermal noise in the PIC leads to the growth. However, Vlasov simulations show wave–wave coupling which stabilises the modes. In this work, our focus is to demonstrate the difference in this behaviour and to thoroughly analyse the spectra and wave generation for the same.