Tech billionaire Elon Musk - the company’s founder and chief executive - announced the news barely a week after launching his first rocket from Nasa’s famous moon pad.
Two people who know each other approached the company about sending them on a week-long flight just beyond the moon, according to Mr Musk.
He would not identify the pair or the price tag, but said they have paid a “significant” deposit and are “very serious” about it.
“Fly me to the moon ... Ok,” he said in a light-hearted tweet following the news conference.
Mr Musk said SpaceX is on track to launch astronauts to the International Space Station for Nasa in mid-2018.
This moon mission would follow about six months later, by the end of the year under the current schedule, using a Dragon crew capsule and a Falcon heavy rocket launched from Nasa’s former moon pad in Florida.
If all goes as planned, it could happen close to the 50th anniversary of Nasa’s first manned flight to the moon, on Apollo 8.
The SpaceX moonshot is designed to be autonomous - unless something goes wrong, Mr Musk said.
“I think they are entering this with their eyes open, knowing that there is some risk here,” he told reporters.
“They’re certainly not naive, and we’ll do everything we can to minimise that risk, but it’s not zero,” said Mr Musk, adding that the pair will receive “extensive” training.
He said he did not have permission to release the passengers’ names, and would not say if they were men, women or even pilots. He would only admit: “It’s nobody from Hollywood.”
The paying passengers would make a long loop around the moon, skimming the lunar surface and then going well beyond, perhaps 300,000 or 400,000 miles. It’s about 240,000 miles to the moon alone, one way.
The mission would not involve a lunar landing.
“This should be a really exciting mission that hopefully gets the world really excited about sending people into deep space again,” Mr Musk said.
In a statement, Nasa commended SpaceX “for reaching higher”. In all, 24 astronauts flew to the moon and 12 walked on its surface from 1969 to 1972.
California-based SpaceX already has a long list of firsts, with its sights ultimately set on Mars. It became the first private company to launch a spacecraft into orbit and safely return it to Earth in 2010, and the first commercial enterprise to fly to the space station in 2012 on a supply mission.
A week ago, SpaceX made its latest delivery from Kennedy Space Centre’s Launch Complex 39A, where the Apollo astronauts flew to the moon and shuttle crews rocketed into orbit.
That will be where the private moon mission will originate as well.