The unnamed passengers have already paid a "significant deposit" to do a Moon mission aboard the company's Dragon spacecraft.
"We are excited to announce that SpaceX has been approached to fly two private citizens on a trip around the Moon late next year," SpaceX CEO Elon Musk said in a statement.
"This presents an opportunity for humans to return to deep space for the first time in 45 years and they will travel faster and further into the solar system than any before them," said Musk.
As part of NASA's Commercial Crew Programme, SpaceX will launch its Crew Dragon (Dragon Version 2) spacecraft to the International Space Station (ISS) this year.
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This first demonstration mission will be in automatic mode, without people on board, the company said.
A subsequent mission with crew is expected to fly in the second quarter of 2018.
"Like the Apollo astronauts before them, these individuals will travel into space carrying the hopes and dreams of all humankind, driven by the universal human spirit of exploration," SpaceX said.
"By also flying privately crewed missions, which NASA has encouraged, long-term costs to the government decline and more flight reliability history is gained, benefiting both government and private missions," the company said.
Once operational Crew Dragon missions are underway for NASA, SpaceX will launch the private mission on a journey to circumnavigate the Moon and return to the Earth.
Lift-off will be from Kennedy Space Center's historic Pad 39A near Cape Canaveral - the same launch pad used by the Apollo programme for its lunar missions.
These missions will build upon that heritage, extending it to deep space mission operations, an important milestone as towards the company's ultimate goal of transporting humans to Mars.
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