The present study investigates the linear spatio-Temporal and weakly nonlinear stability of a pressure-driven two-layer channel flow subjected to a wall-normal temperature gradient commonly encountered in industrial applications. The liquid-liquid interface tension is assumed to be a linearly decreasing function of temperature. The study employs both numerical (pseudo-spectral method) and long-wave approaches. The general linear stability analysis (GLSA) predicts shear-flow and thermocapillary modes that arise due to the imposed pressure and temperature gradients, respectively. The previous stability analyses of the same problem predicted a negligible effect of the pressure-driven flow on the linear stability of the system. However, the GLSA reveals stabilising and destabilising effects of the pressure-driven flow depending on the viscosity ratio , thermal conductivity ratio , interface position and the sign of the imposed temperature gradient . The analysis predicts a range of for given and, which can not be stabilised by the thermocapillarity. The numerically predicted long-wave instability is then captured using the long-wave asymptotic approach. The arguments based on the physical mechanism further successfully explain the role of, the sign of and the interaction between the velocity and temperature perturbations in stabilising/destabilising the flow. The spatio-Temporal analysis reveals the dominance of the spanwise mode in causing the absolutely unstable flow. The weakly nonlinear analysis reveals a subcritical pitchfork bifurcation without shear flow. However, with the shear flow, the streamwise mode undergoes a supercritical Hopf bifurcation. © The Author(s), 2023. Published by Cambridge University Press.