we devote to investigate the quenching phenomenon for a reaction-diffusion system with coupled singular absorption terms, ut=Δu−u−p1v−q1, vt=Δv−u−p2v−q2. The solutions of the system quenches in finite time for any initial data are obtained, and the blow-up of time derivatives at the quenching point is verified. Moreover, under appropriate hypotheses, the criteria to identify the simultaneous and nonsimultaneous quenching are found, and the four kinds of quenching rates for different nonlinear exponent regions are given. Finally, some numerical experiments are performed, which illustrate our results.