The couple would travel about 300,000 to 400,000 miles into space
Planned for late 2018, the unnamed pair have already paid a “significant deposit” for the mission according to SpaceX CEO Elon Musk.
And they will travel aboard the private company’s Crew Dragon capsule which is set for its first unmanned test flight this year.
Mr Musk claimed the co-operation of NASA had made the plan possible, adding the two passengers "will travel faster and further into the solar system than any before them".
The CEO said: "This presents an opportunity for humans to return to deep space for the first time in 45 years.”
They will travel aboard the private company’s Crew Dragon capsule
But he declined to comment on the identity of the two individuals who approached the company last year, only saying that they knew each other and that it’s “nobody from Hollywood”.
He added: "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.
"We expect to conduct health and fitness tests, as well as begin initial training later this year."
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Mr Musk acknowledged the flight would be dangerous but said the customers have “their eyes open” about what could happen and were “very serious” about the flight.
He added: “We’re doing everything we can to minimise that risk, but it’s not zero.
“We would expect to do more than one mission of this nature.”
CEO Elon Musk claimed the co-operation of NASA had made the plan possible
The SpaceX vessel is set for its first unmanned test flight this year
Like the Apollo astronauts before them, these individuals will travel into space carrying the hopes and dreams of all humankind
SpaceX CEO Elon Musk
The journey would take about a week and would see the two space tourists travel about 300,000 to 400,000 miles into space – nearing the moon’s surface without landing before it returned to Earth.
If SpaceX is able to complete the trip before NASA or any other space agency can send astronauts to the moon again, it will be the first lunar mission with humans in 45 years – and could take them further into space than anyone has travelled ever before.
The course would even extend past the record 249,000 miles travelled by the Apollo 13 astronauts in 1970.
The US has not sent astronauts to the Moon since NASA's Apollo missions
The mission will be carried out in 2018
A company statement said: “Other flight teams have also expressed strong interest and we expect more to follow.
“Additional information will be released about the flight teams, contingent upon their approval and confirmation of the health and fitness test results.”
The Crew Dragon capsule which the two will travel in is based on the design currently used to send cargo to the International Space Station, although SpaceX have allowed for upgrades to ensure it is fit for human transport.
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The capsule will launch on top of the California-Based company’s Falcon Heavy rocket, which is scheduled for its first test flight this summer.
The rocket "will be the most powerful vehicle to reach orbit after the Saturn V moon rocket", Mr Musk said.
He added: ”At five million pounds of liftoff thrust, Falcon Heavy is two-thirds the thrust of Saturn V and more than double the thrust of the next largest launch vehicle currently flying."
The US has not sent astronauts to the Moon since NASA's Apollo missions of the 1960s and 1970s.