In this study, we determine the near real-time (NRT) clocks of the Global Positioning System (GPS) and Globalnaya Navigatsionnaya Sputnikovaya Sistema (GLONASS) satellites in the Taiwan RO Process System (TROPS), which is mainly designed for the data processing in both the FORMOSAT-3/COSMIC (F3C) and FORMOSAT-7/COSMIC-2 (F7C2) satellite missions. The accuracy of GNSS clocks defines the quality of the atmospheric excess phase, which is used for the retrieval of bending angle profiles in GNSS radio occultation (RO) observations. The accuracy of the NRT GNSS clocks is assessed by comparing the clock rate, clock stability and clock-induced positioning error on receivers with the final solutions given by the European Space Agency (ESA). Overall, the standard deviations of the clock rates from TROPS agree with those from ESA within 0.05 mm/s over 2304 clock solutions. Additionally, we find that the clock stability of the GPS Block IIF type (3 × 10−13) is an order of magnitude better than that of IIR Block types (3 × 10−12) over a time interval of 30 s. In comparison, the stabilities of GLONASS clocks are approximately 3 × 10−12. We quantify the NRT clock error on the receiver positioning by using the precise point positioning technique obtained from the Bernese GNSS software. The 3-dimensional clock-induced positioning error is approximately 3.3, 3.2 and 0.9 cm for station AUCK and 6.9, 6.3 and 3.1 cm for station NRC1 for the GPS-only, GLONASS-only and GPS + GLONASS cases, respectively. For GNSS-RO applications, the bending angle profiles derived using TROPS GPS clocks agree with the COSMIC Data Analysis and Archive Center products to within 0.01–1.00 μrad. However, this is not the case for the GLONASS clock, because the GLONASS clock-induced errors on the RO profile are 10–100 times greater than those induced by the GPS clock. This suggests that different weightings should be used for RO applications, such as data assimilation, when different satellite clocks are involved in GNSS-RO retrievals. This study serves as a reference for assessing the impact of GNSS clocks on both GNSS-POD (precise orbit determination) and GNSS-RO in preparation for the F7C2 satellite mission.