In this paper, a fully discrete finite volume element (FVE) scheme is proposed to treat the linear multi-term time fractional reaction-diffusion models with variable coefficients on triangular grids, where the Caputo fractional derivative of time variable is approximated by the $L1$ formula. The existence and uniqueness of solution for the proposed scheme is proved based on the matrix theories, the stability results are derived in detail by using a multi-term fractional Gronwall inequality, and the optimal error estimates under the $L^2(\mathrm{\Omega })$ and $H^1(\mathrm{\Omega })$ norms are also obtained. Owing to the variable coefficients in the models, a special technique should be used to analyze the stability and error estimates. Finally, three numerical examples with different fractional derivative terms are given to validate the feasibility and effectiveness.