In this paper, we establish a susceptible-infected-removed (SIR) rumor spreading model based on the influence of rumor-refuting mechanism and time delay on internet rumor spreading. The threshold R₀ of rumor spreading is obtained by using the spectral radius method of regenerative matrix; the conditions for the existence of rumor prevailing equilibrium point are given according to the quadratic function characteristics; the local stability of rumor-free equilibrium point and rumor prevailing equilibrium point are established by using eigenvalue theory and Routh-Hurwitz criterion; and the criterion for the occurrence of Hopf bifurcation is also established. The numerical simulation results show that the information about refuting rumors, released by the government and the media, can accelerate the convergence rate of rumors and reduce the maximum density of rumor-spreaders.