We analyze the nonequilibrium thermodynamic pressure which is obtained by differentiation of a generalized nonequilibrium entropy. It is proposed that this nonequilibrium thermodynamic pressure, π, is related to the pressure tensor P in the presence of a homogeneous heat flux q→ through P=πU+αq→ q→, with α given by the condition trP=3p, p being the local-equilibrium pressure. A simple situation illustrating the physical difference between π and p is presented. The nonequilibrium entropy, pressure, and pressure tensor are also analyzed from a microscopic point of view in several systems. © 1995 The American Physical Society.