This paper addresses the precise clock estimation problem on the lunar surface, especially at the South Pole region. The problem is particularly challenging because the moon is about 380.000 km away from the earth and there are currently no systems in the moon vicinity that can be directly utilized to estimate the receiver clock to be located on the lunar surface. We tackle this problem by assuming our lunar navigation satellite system (LNSS) under planning. The LNSS is a satellite constellation system to provide the communication, positioning, navigation, and timing (CPNT) service at the South Pole region. The LNSS is a GPS-like system and the LNSS satellites will be designed to broadcast the navigation signals in the same way as the GPS. We investigate the achievable clock estimation accuracy at the South Pole by using the LNSS. Our clock estimation is based on the filtering technique that processes the LNSS pseudo range and carrier phase measurements and the simulation results showed that a nanosecond level ? 10?9 estimation accuracy is continuously expected by using our filtering approach.