In this work, we develop a weak Galerkin finite element scheme for the Stokes interface problems with curved interfaces. The conventional numerical schemes rely on the use of straight segments to approximate the curved interfaces and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. Theoretical analysis and a series of numerical experiments demonstrate that errors reach the optimal convergence orders under both the energy norm and the L2 norm.