Convective flows of fluids under action of a cocurrent gas flux and gas related evaporation effects are actively investigated nowadays analytically, numerically and experimentally. The problem of mathematical modeling of convective processes with evaporation is extremely complicated. Formulation of boundary conditions on the interface between liquid and gas-vapor mixture is a crucial aspect here. In this paper, generalized kinematic, dynamic and energy conditions on the thermocapillary interface are presented. The interface conditions are formulated with consideration of transfer coefficient temperature dependence under a hypothesis that free surface energy is identified with a surface tension coefficient, Stokes’ law is used for incompressible fluid, heat and vapor fluxes are derived from Fourier and Fick laws, and latent heat of evaporation is a jump of inner potential energy. Continuity of temperature and tangential velocities on the interface is assumed, and evaporation mass flux is obtained from kinetic theory.