NASA requested $363 million for fiscal year 2020 for the Advanced Cislunar and Surface Capabilities (ACSC) program, more than double what the space agency sought last year. More importantly, NASA estimates the budget for the program would grow to $2.4 billion by fiscal year 2024, as the agency begins to award companies with development contracts.

In a potential boon for private space companies like Elon Musk's SpaceX and Jeff Bezos' Blue Origin, NASA is asking Congress for billions of dollars over the next five years to speed the development spacecraft capable of flying humans to and from the moon.

The huge increase in this year's budget request for ACSC is the result of an internal battle at NASA, a person familiar with the situation told CNBC on Monday. This year's introduction of a competitive bidding process means that companies would build their own spacecraft, with help from competitive NASA funding awards, rather than build a NASA-specific spacecraft.

NASA didn't respond to CNBC requests for comment.

After the establishment of the National Space Council, President Donald Trump's administration looked to work with the growing private space sector on multiple NASA programs. A main directive was to create a sustained U.S. presence on the moon. NASA would first send robots to the surface without crews, later following up with astronauts.

Recently appointed NASA administrator Jim Bridenstine supported a White House plan last year to make ACSC a competitive process between companies, two people familiar with the matter told CNBC. But some NASA leaders rejected the idea, wanting to stick with the development of an in-house lunar lander project, one of the people said.

Known as the "Flexible Lunar Explorer" (or FLEx) lander concept, some at NASA wanted a system built for the agency, rather than according to the companies building the spacecraft. This is NASA's traditional method of awarding contracts, exemplified by the Boeing-built Space Launch System (SLS) rocket – which is now years behind and racking up billions of dollars in cost overruns.

With delays in the SLS program, privately-built rockets are being increasingly considered to launch NASA missions. And now, despite internal hesitancy, NASA is considering commercial alternatives for getting astronauts to the moon's surface. The agency has spoken to companies about enabling "regular access to the lunar surface," NASA said in its budget request earlier this month.

The increased funding for ACSC also matches with statements Vice President Mike Pence made on Tuesday at a National Space Council meeting. Pence said it is the White House policy "to return American astronauts to the moon within the next five years." Although he said the SLS rocket must be accelerated to do so, he urged NASA to reach the moon "by any means necessary."

"We're not committed to any one contractor. If our current contractors can't meet this objective, then we'll find ones that will," Pence said. "If American industry can provide critical commercial services without government development then we'll buy them."

Musk welcomed Pence's comments, saying in a tweet that "it would be so inspiring for humanity to see humanity return to the moon!"