According to SpaceX, only 24 humans have been to the moon in history. Back in February 2017, the company announced that two people had signed up for a week-long around the moon mission with a launch date expected late 2018.
"Find out who's flying and why on Monday, September 17", the tweet said.
SpaceX originally planned to send two paying passengers on a flight around the moon this year using its Falcon Heavy rocket and a Dragon capsule.
However the passengers were never named and the flight has yet to materialise.
SpaceX now has a $1.6 billion contract with NASA to supply the astronauts living at the ISS, via regular cargo trips on its Dragon spaceship, launched aboard a Falcon 9 rocket.
US President Donald Trump has championed plans to put boots on the Moon again, as NASA works on building a lunar gateway that would serve as a launching point for missions heading even further into deep space, such as asteroids or Mars.
Musk described a slightly scaled-down 348-foot-tall rocket in September 2017 and announced that the private space company aimed to launch two cargo missions to Mars in 2022.