Abstract Constrained fluid flow and conjugate heat transfer in microchannel heat sinks (MCHS) with converging channels are investigated using the finite volume method (FVM) in the laminar regime. The maximum pressure of the MCHS loop is assumed to be limited due to constructional or operational conditions. Results show that the Poiseuille number increases with increased tapering, while the required pumping power decreases. Meanwhile, the Nusselt number increases with tapering as well as the convection heat transfer coefficient. The MCHS having the optimum heat transfer performance is found to have a width-tapered ratio equal to 0.5. For this tapering configuration and at the maximum pressure constraint of 3000 Pa, the pumping power reduces by a factor of 4 while the overall heat removal rate is kept fixed in comparison with a straight channel.