The mission will be the first by the US space agency since its Apollo mission, the report said.
The bid by Musk, who also leads Tesla Inc, beat one from Amazon.com Chief Executive Jeff Bezos, had partnered with Lockheed Martin Corp, Northrop Grumman Corp and Draper, according to the report. Bezos also owns the Washington Post.SpaceX earlier this week announced it had raised about $1.16 billion in equity financing.