This paper investigates the vibration behavior of fluid conveying viscoelastic pipe rested on non-uniform elastic Winkler foundation. The Kelvin-Voigt model is employed to consider the viscoelastic behavior of the pipe. Using the Galerkin’s method, the eigenvalue problem for the simply supported fluid conveying viscoelastic pipe is extracted. The effects of the fluid velocity, the viscoelastic constants and the foundation parameters on the complex eigenvalues and the divergence and the flutter instability of the fluid conveying viscoelastic pipe are studied and discussed. It is found that including the viscoelastic behavior to the pipe material alters the trend of the instability of the fluid conveying pipe, i.e., the first and the second modes divergence and the coupled mode flutter for the elastic pipe change to the first mode divergence, the second mode flutter and the second mode divergence for the viscoelastic pipe, respectively. The structural damping causes the velocity of the divergence instability at the higher modes to be increased. Also, because the viscoelasticity of the pipe affects the different vibration modes in different manner, therefore, the pipe dose not exhibit a coupled-mode flutter. Moreover, the non-uniformity of the foundation stiffness alters the first divergence velocity. The results are verified through comparing them with those reported in the literature.