Convective heat transfer and effect of nonlinear wall slip are studied analytically in concentric microannulus for viscoelastic fluids obeying the Giesekus constitutive equation. Laminar, thermally, and hydrodynamically fully developed flow is considered. A nonlinear Navier model with nonzero slip critical shear stress is employed for the slip equation at both walls. Critical shear stress will cause three slip flow regimes: no slip condition, slip only at the inner wall, and slip at both walls. Thermal boundary conditions are assumed to be peripherally and axially constant fluxes at the walls. Governing equations are solved to obtain temperature profiles and Nusselt number and effects of slip parameters, elasticity, and Brinkman number are discussed. Two regimes are compared when slip occurs at both walls or only at the inner wall. The results indicate that by increasing slip effect and elasticity, heat transfer between wall and fluid is enhanced, but it decreases by increasing Brinkman number. In the case where the heat flux is dominant in the outer wall, the inner wall Nusselt curve shows a singularity for a critical Brinkman number because at this Brinkman number the bulk temperature will be equal to the wall temperature.