We report on the temperature dependence of the critical current (Ic) nearTcfor two Y‐Ba‐Cu‐O thin films having zero‐dissipation temperature (Tc) of 81 K and a Y‐Ba‐Cu‐O thin film havingTcof 71 K. At temperatures higher than 0.9Tc, the critical current in a voltage criterion of 1 &mgr;V/cm appears proportional to (1−T/Tc)nwithn≊2, andn≊1.5 in a 10 &mgr;V/cm criterion, exhibiting the dependence ofnon the critical current criteria. This observation is in contrast with the report ofn=1.5 by Ogaleetal. [Phys. Rev. B36, 7210 (1987)], Yuanetal. [J. Appl. Phys.64, 4091 (1988)], and Horngetal. [Phys. Rev. B39, 9628 (1989)], implying thatn=2, the value for the superconductor‐normal metal‐superconductor tunneling junction of BCS superconductors nearTcmust be taken into consideration to understand the intergrain tunneling process in polycrystalline highTcthin films.