Abstract: Thermocapillary migration of a droplet placed on a non-uniformly heated solid surface appears in a variety of practical applications, such as microfluidics, inkjet printing, et al. The flow stability analysis is crucial for the precise control of droplet migration. In the present work, the convective instability in thermocapillary migration of a wall-attached viscoelastic droplet is examined by linear stability analysis. The relation between the critical Marangoni number (Ma_\rm c) and the elastic number is obtained at different Prandtl numbers (Pr). The flow fields and energy mechanisms of preferred modes are analyzed. The results show that more kinds of preferred modes are excited by the elasticity. The preferred modes at small Pr are the oblique and streamwise waves, while those at moderate and high Pr are oblique waves and spanwise stationary modes. The strong elasticity significantly reduces the Ma_\rm c, while the weak elasticity slightly enhances the flow stability. Ma_\rm c increases with Pr at moderate Pr. For the oblique wave, the amplitude of perturbation temperature may appear in the middle region of flow field, while the amplitudes of other two modes only exist on the free surface. The distribution of streamlines is almost symmetric at high Pr. The energy analysis shows that the work done by the basic flow changes from positive to negative when the elastic number increases. The work done by the perturbation stress may either dissipate or provide energy at small Pr, while the work done by the basic flow is negligible at high Pr. For the downstream streamwise wave, the perturbation velocity and the work done by perturbation stress fluctuate several times in the vertical direction. Comparing the droplet migration with thermocapillary liquid layers, it can be found that due to the differences of basic flow and boundary conditions, there are quite different between their preferred modes and perturbation flow fields.