The linear stability analysis of the onset of double-diffusive convection in a Poiseuille flow system is investigated. In addition, a volumetric uniform internal heat source is taken into account. In this problem, the horizontal fluid channel is bounded by two plates which are isothermal and isosolutal. The governing parameters are thermal Rayleigh number RaT, solutal Rayleigh number Ras, internal heat source parameter RaI, Prandtl number Pr, and Reynolds number Re. The eigenvalue problem arising from the linear perturbed system of equations is solved numerically using the Chebyshev–Tau method coupled with the QZ algorithm. It is found that the positive solutal Rayleigh number Ras destabilizes the system. Furthermore, it is observed that an increase in the Prandtl number Pr stabilizes the system. Additionally, at Ras = −60, the critical values of the thermal Rayleigh number Rac decreases with R=Re cos ϕ up 2; and increases with R beyond R=2.