We performed direct numerical simulation of fully developed turbulent velocity and temperature fields in a flume, for Reynolds number, based on the wall shear velocity and the height of the flume, Re=171 and Prandtl numbers Pr=1.0 and Pr=5.4. To elucidate exactly the role of the wall boundary condition for passive scalar, the system considered was the flow at constant properties of the fluid. Two types of thermal wall boundary conditions (BCs) for the dimensionless temperature equation were studied: isothermal wall boundary condition—H1, and isoflux wall boundary condition—H2. The profile of the mean temperature was not affected by the type of BC. However, the type of BC has a profound effect on the statistics of the temperature fluctuations in the near-wall region y+<10. Comparison of near-wall statistics of temperature fluctuations shows that at Pr=1 the buffer part of the turbulent boundary layer significantly influences the scalar transfer in the conductive sublayer, whereas at Pr=5.4 the near-wall temperature field may be associated with predominant motion in the viscous sublayer.