In this paper, we consider the problem of constrained fast source seeking using two nonholonomic wheeled robots with velocity limitations. To achieve maximum converging speed to the source, we propose a dual-module control approach. In the formation control module, we design the angular velocities of the two robots to keep them in a formation. In the source seeking module, a sliding-mode-like control algorithm is developed to generate the bounded linear velocities so that the moving directions of the two robots are aligned with the field gradient direction as quickly as possible. Only the instantaneous measurements of the field but not the gradients are used in the controller design. Compared to existing approaches, our method makes full use of the maximum velocity limitations of the nonholonomic robots to achieve fast overall converging speed using only two robots. Comparative simulation and experimental results are presented, demonstrating the superior performance of the proposed control strategy over the existing ones in terms of the seeking (converging) speed.