The new-generation global navigation satellite system (GNSS) satellites are committed to providing multi-frequency signals for establishing improved positioning, navigation and timing (PNT) services, including GPS Block IIF satellites, Galileo satellites, GLONASS-K satellites and BDS-2/BDS-3 satellites. The joint usage of multi-frequency signals draws increasing attention from the GNSS community. However, the satellite clocks provided by the international GNSS service (IGS) are a lumped term of frequency-independent physical clock errors, and frequency-dependent code and carrier phase hardware delays at the satellite. The current precise satellite clock products are generated based on specific observations, such as L1/L2 ionospheric-free (IF) combined code and carrier phase observations for GPS. The inconsistency among signal-dependent biases within a satellite results in the inadequacy of the current L1/L2 IF satellite clock products for the GPS precise point positioning (PPP) involving L5 signal. This study focuses on the consistent use of satellite clocks for GPS triple-frequency PPP. We can first estimate the inter-frequency clock bias (IFCB), and then, convert the L1/L2 IF satellite clocks into the desired satellite clocks for triple-frequency PPP by combining them with the estimated IFCBs. However, there are various types of triple-frequency PPP models, including the PPP model with L1/L2 and L1/L5 dual-frequency IF combinations (IF-PPP1), the PPP model using the triple-frequency observations integrated into a single IF combination (IF-PPP2), and the PPP model based on the uncombined (UC) observations (UC-PPP). In addition, for the IF-PPP2, there is an infinite number of triple-frequency IF combinations with only two conditions, namely removing the first-order ionospheric delay and keeping the geometric distance unchanged. Therefore, we should estimate various types of IFCBs to implement the triple-frequency PPP processing, according to the differentrequirements of users. Definitely, this will increase the workload of satellite clock providers and multi-frequency PPP users. A unified IFCB estimation approach, which can be applied to various triple-frequency PPP models, is needed. If we can know the relationship among different types of IFCBs, all of them can be obtained by converting only a set of estimated IFCBs, such as L1/L5 IF IFCBs (difference between L1/L5 and L1/L2 IF satellite clocks). In this study, we have rigorously derived the mathematical conversion formula. Thus, the satellite clocks for various types of triple-frequency PPP models can be derived by combining a set of current precise satellite clock products with only a set of IFCBs. The multi-frequency integration becomes easy. Although our investigation is based on GPS data, the relevant theory and approach can also be applied to BDS.