In this paper, a high order finite volume scheme for solving the non-conservative convection equations on the unstructured grids is proposed. It is found that when the non-conservative convection equations are rewritten into the conservative form with additional source term, the direct application of the finite volume scheme using high order reconstruction will produce numerical instability. To solve this problem, we propose in the present paper to solve the integral form of the non-conservative convection equations. To account for the upwinding effect, a convective reconstruction technique is proposed. The proposed method is applied to solve a linear advection equation and the eikonal equation in time dependent non-conservative form. An artificial viscosity term is added to handle the singularity of the equation. The numerical results show that the proposed numerical scheme can achieve high order accuracy and is very robust.