According to Musk, SpaceX was approached by the two individuals who expressed interest in the mission, which is expected to skim the moon and carry them into deep space on the company's Dragon Capsule and Falcon Heavy rocket after launch from Kennedy Space Center in Florida.
A press release from SpaceX noted that the pair "have already paid a significant deposit to do a moon mission." Musk declined to reveal the clients' identities. The company said that additional information about the team will be released following approval, health and fitness tests.
SpaceX is planning to send an unmanned Dragon craft to the International Space Station later in the year before launching a manned mission in 2018. The company has contracted by NASA to continue missions to the space station at a rate of four per year.
The trip around the moon is scheduled to take place after SpaceX has completed successful manned missions for NASA.
ABC's Gina Sunseri contributed to this report.