It is well known that the plane channel flows of UCM and Oldroyd-B fluids are linearly stable at low Reynolds numbers. However, an elastic instability is detected by modal analysis in the thermocapillary convection with straight streamlines. The instability appears when the Weissenberg number is in the order of 1. When the Reynolds number is small enough, the critical Weissenberg number is almost a constant, which only depends on the Marangoni number. The unstable waves include a wide range of wave number and can be streamwise and oblique. The free surface with thermocapillary effect is crucial for the instability, as it induces the normal stress in the basic flow and reduces the constraint on perturbations.