After someone responded to the tweet by jokingly suggesting that Musk himself will be the passenger, the SpaceX CEO offered a hint about the person who is traveling, responding with a Japanese flag emoji.
The lunar mission will take place on the company's planned Big Falcon Rocket, a massive launch vehicle SpaceX is developing to make the trip to the moon and Mars.
SpaceX is set to send a private passenger on a spaceflight around the moon soon, as it's now signed up its first customer for the trip.
SpaceX gave no further details, but said more information would follow on Monday.
As per a tweet on Thursday, the company has signed its first private passenger to fly on its BFR launch vehicle, in what would be "an important step toward enabling access for everyday people who dream of traveling to space".
Other non-astronauts that have flown into space include inventor Gregory Olsen and multimillionaire Dennis Tito, the first tourist in space, after he paid $20 million to spend 8 days on the ISS in April 2001. But Musk has broad ambitions for SpaceX and the BFR.
In an interview in March, Musk said the ship was now being built, adding "I think we'll probably be able to do short flights, short sort of up-and-down flights, probably sometime in the first half of next year".
The 31-engine BFR is part of the company's grand plan for travelling between planets, and will replace its current suite of rockets like the Falcon 9 and Falcon Heavy.
The private space company, led by billionaire Elon Musk, has previously announced plans to send two people into lunar orbit, but has never revealed who they may be. The window for launching to Mars occurs every two years.