The present study has numerically investigated two-dimensional laminar flow over a steadily rotating circular cylinder with a uniform planar shear, where the free-stream velocity varies linearly across the cylinder. It aims to find the combined effect of rotation and shear on the flow. Numerical simulations using the immersed boundary method are performed for the ranges of $-2.5{le}alpha{le}2.5$ and $0{le}K{le}0.2$ at a fixed Reynolds number of Re=100, where a and K are respectively the dimensionless rotational speed and velocity gradient. Results show that the positive shear, with the upper side having the higher free-stream velocity than the lower one, favors the effect of the counter-clockwise rotation $(alpha<0)$ but countervails that of the clockwise rotation $(alpha>0)$. Accordingly, the absolute critical rotational speed, below which vortex shedding occurs, decreases with increasing K for $(alpha>0)$, but increases for $alpha>0$. The vortex shedding frequency increases with increasing alpha (including the negative) and the variation becomes steeper with increasing K. The mean lift slightly decreases with increasing K regardless of the rotational direction. However, the mean drag and the amplitudes of the lift- and drag-fluctuations strongly depend on the direction. They all decrease with increasing K for $alpha>0$, but increase for $alpha<0$. Flow statistics as well as instantaneous flow folds are presented to identify the characteristics of the flow and then to understand the underlying mechanism.