Nonlinear shallow water equations with Coriolis forces in low Froude number regimes are widely used in the modeling free-surface flows in rivers and coastal areas. Explicit numerical schemes for the equations require that the time step must satisfy dt=O(epsilon*dx), where epsilon is a small parameter called as the Froude number. As the Froude number becomes smaller, this leads to an increasingly large computational cost. To avoid this issue, we  present an asymptotic preserving IMEX RKDG method for solving the equations. In the presented method, the equations are first splitted into two parts, then the linear stiff parts of the equations are solved implicitly and the nonlinear non-stiff parts are solved explicitly. Moreover, the presented scheme is well-balanced for the equations over variable bottom topography.. Finally, several numerical tests are carried out to validate the performance of our proposed schemes.