A study of heat transfer through a high temperature superconducting (HTSC) current lead in both, resistive and superconducting mode, has been performed. The set of dimensionless governing parameters was defined, and the physical meaning of different parameters elaborated. An optimal selection of thermal design parameters was suggested. The selection of parameters has been based on : (i) the numerical solution of a set of nonlinear governing differential equations; and (ii) an estimation of the irreversibility induced by heat transfer and Joule heating phenomena. The results were discussed for a YBa2Cu3O7-delta (YBCO) compound system. The case study has been based on a cryogenic current lead model that includes the nonlinearities as follows : (i) variable thermal conductivity; (ii) variable electrical resistivity (in resistive mode); and (iii) Variable heat transfer coefficient. The results obtained correspond to the temperature range between 4.2 and 77 K. It was shown that a local variation in heat transfer coefficient does not influence the selection of optimal design parameters.