To investigate the normal-state magnetic properties of UTe2 under pressure, we perform Te125 nuclear magnetic resonance (NMR) measurements up to 2 GPa. Below 1.2 GPa, the b-axis NMR Knight shift shows a broad maximum at the so-called Tχmax on cooling, which is consistent with the magnetization measurement under pressure. Tχmax decreases with increasing pressure and disappears at the critical pressure Pc = 1.7 GPa, above which superconductivity is destroyed. This tendency is also observed in the temperature dependence of the nuclear spin-lattice relaxation rate 1/T1. At low pressures, 1/T1 shows a conventional Fermi-liquid behavior (1/T1T=const) at low temperatures, indicating the formation of the heavy-fermion state. Above Pc, 1/T1T follows a 1/T behavior without any crossover to the heavy-fermion state down to the lowest temperature (∼3 K). In addition, the NMR signals disappear below 3 K, due to the influence of the magnetically ordered moments. From the pressure dependence of the Tχmax and Knight shift, it was found that the Fermi surface character is abruptly changed at Pc, and that superconductivity is observed only in the heavy-fermion state.