The results of experiments [C. Y. Han &etal;, Low Temp. Phys.24, 305 (1998)] on the influence of quasi-hydrostatic pressurePon the superconducting transition temperatureTcin polycrystalline samples of cuprate compoundsTl2Ba2Ca2Cu3Ox(Tl-2223) with the initial value ofTc=129&hthinsp;Kand the maximum valueTc=255.4&hthinsp;KatP=4.3&hthinsp;GPaare considered. A hypothesis on the existence of two mechanisms of the effect of pressure onTcis proposed on the basis of an analysis of the nonmonotonic dependence ofTconP.The first mechanism operating in the pressure rangeP<3&hthinsp;GPais connected with the anisotropic deformation of the crystal lattice of individual grains (crystallites) in the plane of cuprate layersCuO2and with a change in the oxygen doping of layers. The second mechanism operating in the pressure rangeP>3&hthinsp;GPais due to hydrostatic compression of the ionic lattice and an increase in the bulk concentration of charge carriers (holes) in view of the electroneutrality condition. ©1998 American Institute of Physics.