Bezos' space company Blue Origin has teamed up with aerospace giants Lockheed Martin, Northrop Grumman and Draper to bid for the landing system.
The White House has set the ambitious goal of sending a man and a woman to the lunar South Pole within five years.
Bezos outlined the plan at a meeting in Washington DC.
The Amazon founder called the partnership "a national team for a national priority".
Nasa had originally planned to mount the Moon return mission in 2028. But earlier this year, Vice President Mike Pence announced the administration's plan to accelerate that timeline by four years.
"Ladies and gentlemen, that's just not good enough," Mr Pence said about the original schedule when announcing the 2024 target in March.
The programme has been named Artemis, after the sister of Apollo.
Blue Origin, a relatively new entrant in the human spaceflight sector, will lead the team, which is composed of companies with decades of experience building hardware for Nasa.
"We recognise that this project and the time frame that the nation is calling for is ambitious, very ambitious," Brent Sherwood, Blue Origin's vice president of advanced development programs was reported as saying by the Washington Post.