In this paper, a size-dependent viscoelastic pipe model is developed to investigate the size effects on flutter and divergence instability of functionally graded viscoelastic nanotubes conveying fluid. The nonlocal strain gradient theory and the Kelvin-Voigt model are used to consider the significance of nonlocal field, strain gradient field, and viscoelastic damping effects. The dimensionless equation of transverse motion and related classical and non-classical boundary conditions are derived using the variational approach. The partial differential equations are discretized to a system of ordinary differential equations by the use of Galerkin's method. The frequency equation is obtained as a function of dimensionless flow velocity, small-scale parameters, damping coefficient, and power-law parameter. Numerical results are presented to study the dynamical behavior of the system and are compared with experimental and theoretical results reported by other researchers. Coupled and single mode fluttering related to higher vibration modes of fluid-conveying nanotubes supported at both ends are studied for the first time. It is found that coupled mode fluttering can be seen for different vibration modes by increasing the flow velocity in the absence of structural damping. Structural damping changes the dynamical behavior of the system, in which by increasing the flow velocity, single mode fluttering occurs instead of coupled mode fluttering. In addition, the presence of structural damping increases the critical flow velocity and, as a result, increases the stability of the system. The results also show that increasing the nonlocal parameter will have a stiffness-softening effect, while increasing the strain gradient length scale has an opposing effect.