Motivated by the fact that a leaking pipe can lose or gain energy from the leaking flow, this study attempts to explore the nonconservative leaking flow effect on the dynamic stability of a simply supported pipe with a constant velocity leakage. It employs a two-dimensional nonlinear longitudinal and lateral coupling model, and the leakage effect is accounted for by virtual work due to virtual momentum transport at the leaking point. The equations of motion are solved by Galerkin-based multimode approach and the Houbolt's finite difference time integration. It demonstrates that when there is a leaking flow, a stable pipe can be refined or destabilized via a static pitchfork bifurcation, and a buckling pipe can be stabilized or deteriorated into a worse divergence condition. The critical leaking flow velocities and the excited buckling modes depend on the leaking fluid mass and the leak's position. This study may provide some insights to assist the leak detection system (LDS) of a pipe transporting high-pressure oil or gas in modern engineering.