We report on the temperature dependence of the critical current (Ic) near Tc for two Y‐Ba‐Cu‐O thin films having zero‐dissipation temperature (Tc) of 81 K and a Y‐Ba‐Cu‐O thin film having Tc of 71 K. At temperatures higher than 0.9Tc, the critical current in a voltage criterion of 1 μV/cm appears proportional to (1−T/Tc)n with n≊2, and n≊1.5 in a 10 μV/cm criterion, exhibiting the dependence of n on the critical current criteria. This observation is in contrast with the report of n=1.5 by Ogale et al. [Phys. Rev. B 36, 7210 (1987)], Yuan et al. [J. Appl. Phys. 64, 4091 (1988)], and Horng et al. [Phys. Rev. B 39, 9628 (1989)], implying that n=2, the value for the superconductor‐normal metal‐superconductor tunneling junction of BCS superconductors near Tc must be taken into consideration to understand the intergrain tunneling process in polycrystalline high Tc thin films.