If all goes well, this unnamed passenger will be the first person to visit the moon since NASA's Apollo mission in 1972.
SpaceX gave no further details, but said more information would follow on Monday.
On Twitter, Musk's space firm wrote: 'SpaceX has signed the world's first private passenger to fly around the Moon aboard our BFR launch vehicle-an important step toward enabling access for everyday people who dream of travelling to space. Speculation swirled following the tweet, leading some to believe it might be SoftBank CEO Masayoshi Son.
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He said: "My thing about my daughter right now is that I think she is terrified". 'I can't believe this - so cool!' one wrote. And earlier this year, the former USA actress wore another chic suit during a royal function with her husband Harry.
However, it was not clear whether the BFR tourism mission has any link to an announcement SpaceX made in February 2017, the CNN reported. By the early 2020s, SpaceX hopes the BFR will replace all of its existing rockets and spacecraft. But Musk has broad ambitions for SpaceX and the BFR. Its design was first presented almost a year ago, and the concept boasted a single system that included a booster and a ship that could travel to the moon and Mars.
In 2017, Space X revealed that two "private citizens" had paid a "significant deposit" to book themselves on a journey around the moon.
Shortly after his successful Falcon Heavy Launch, Musk said he expected a "full-scale test" of the massive rocket within the next few years.
The names and identities of those two tourists - and how much they meant to pay - were never revealed.