In this work we investigate the role played by viscoelasticity on the thermocapillary motion of a deformable Newtonian droplet embedded in an immiscible, otherwise quiescent non-Newtonian fluid. We consider a regime in which inertia and convective transport of energy are both negligible (represented by the limit condition of vanishingly small Reynolds and Marangoni numbers) and free from gravitational effects. A constant temperature gradient is maintained by keeping two opposite sides of the computational domain at different temperatures. Consequently the droplet experiences a motion driven by the mismatch of interfacial stresses induced by the non-uniform temperature distribution on its boundary. The departures from the Newtonian behaviour are quantified via the "thermal" Deborah number, De(T) and are accounted for by adopting either the Oldroyd-B model, for relatively small De(T), or the FENE-CR constitutive law for a larger range of De(T). In addition, the effects of model parameters, such as the concentration parameter c = 1 - beta (where beta is the viscoelastic viscosity ratio), or the extensibility parameter, L-2, have been studied numerically using a hybrid volume of fluid-level set method. The numerical results show that the steady-state droplet velocity behaves as a monotonically decreasing function of De(T), whilst its shape deforms prolately. For increasing values of De(T), the viscoelastic stresses show the tendency to be concentrated near the rear stagnation point, contributing to an increase in its local interface curvature.