This article investigates the effect of the selection of enrichment functions on the formulation of the Generalized Finite Element Method (GFEM) for the solutions of transient heat conduction problems. We present the study of an a-posteriori error estimate with the aim to show it as a reliable tool for the selection of enrichment functions to efficiently capture the sharp thermal gradients of the solutions. Problems in two- and three-dimensional domains are considered to demonstrate the robustness of the proposed error estimate. Numerical experiments consider two different types of enrichment functions that mimic the solution behaviour and capture the time-varying thermal gradients. The presented study shows that the error estimate is independent of the heuristically selected enrichment functions and can be used for any type of enrichment functions. It is concluded that the proposed error estimate efficiently reflects the errors in the GFEM solutions for both types of enrichment functions and can be used as an effective tool for the selection of more suitable enrichment functions that produce lower errors under the considered thermal conditions.