In the present paper a role of the smallest diffusive scales of a passive scalar field in the near-wall turbulent flow was examined with pseudo-spectral numerical simulations. Temperature fields were analyzed at friction Reynolds number Reτ = 170.8 and at Prandtl number, Pr = 5.4. Results of direct numerical simulation (DNS) were compared with the under-resolved simulation where the velocity field was still resolved with the DNS accuracy, while a coarser grid was used to describe the temperature field. Since the smallest temperature scales remained unresolved in this simulation, an appropriate spectral turbulent thermal diffusivity was applied to avoid pileup at higher wave numbers. In spite of coarser numerical grid, the temperature field is still highly correlated with the DNS results, and thus point to practically negligible role of the diffusive temperature scales on the macroscopic behavior of the turbulent heat transfer.