The present study has numerically investigated two-dimensional laminar flow over a circular cylinder with a uniform planar shear, where the free-stream velocity varies linearly across the cylinder. Numerical simulations using the immersed boundary method are performed for the ranges of $50{le}Re{le}160,;K{le}0.2$, and B=0.1 and 0.05 where Re, K and B are the Reynolds number, the non-dimensionalized velocity gradient and the blockage ratio, respectively. Results show that the flow depends significantly on B as well as Re and K. It is found, especially, that the blockage effect accounts for some causes of apparent discrepancies among previous studies on the flow. With increasing K, the vortex shedding frequency and the mean drag stay nearly constant or slightly decrease whereas the mean lift, acting from the higher-velocity side to the lower, increases linearly. Flow statistics as well as instantaneous flow fields are presented to identify the characteristics of the flow and then to understand the underlying mechanism.