SpaceX will use the Falcon Heavy rocket, which was developed with internal funding.
"I think there's a market for one or two of these per year", he said, estimating that space tourist fares charged by SpaceX could eventually contribute 10 to 20 percent of the company's revenue.
According to the Los Angeles Times, Musk said in the teleconference that his company's current contract with NASA would take precedent over the commercial flight should the administration choose to request the two seats on the Dragon capsule's flight around the moon.
That trajectory would be similar to NASA's 1968 Apollo 8 mission beyond the moon and back.
Musk said he does not have permission to release the passengers' names, and he was hesitant to even say if they were men, women or even pilots.
He would not say how much it would cost. There are at least two people with the means and the interest to go out to space, and it does not include anyone from the Hollywood elite.
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The moon is about 240,000 miles away from the Earth, and the travelers will do a flyby of the lunar surface before returning to Earth. This first demonstration mission will be in automatic mode, without people on board. The space agency learned of his plan at the same time as reporters. Lift-off will be from Kennedy Space Center's historic Pad 39A near Cape Canaveral - the same launch pad used by the Apollo program for its lunar missions.
NASA has congratulated SpaceX on "reaching higher". He said there will be ample time to test both the spacecraft and the rocket, before the moon mission.
SpaceX has seen some recent setbacks, including multiple delays launching and docking a capsule on the International Space Station earlier this month.
Lori Garver, who served as deputy administrator of NASA during the Obama administration, told the Washington Post that she was skeptical that SpaceX could pull off such a flight by the end of next year, but said it would be "fantastic" if they could do so by 2020.
Retired NASA astronaut Scott Kelly, who will celebrate his homecoming this week from a one-year space mission, was quick to tweet: "It's been nearly a year".
"A little bit more than the cost of a crewed mission to the space station would be", he said. His ultimate goal is to establish a human settlement on Mars.