They added that the White House also was considering former NASA astronaut and retired Air Force Col. Pamela Melroy to serve as deputy administrator.
If confirmed, Nelson, 78, would replace acting Administrator Steve Jurczyk, who took on the role upon Biden's inauguration on Jan. 20.
Nelson served three terms in the Senate as a Democrat representing Florida. He lost his race for a fourth term to Rick Scott, a Republican, in 2018. Former NASA Administrator Jim Bridenstine -- a pick of former President Donald Trump -- appointed Nelson to NASA's Advisory Council in 2019.
Nelson was the second sitting member of Congress to go into space, traveling on the space shuttle Columbia to orbit the earth for six days in 1986. The then-representative served as a payload specialist on the flight.
"I wish I could learn everything about the space transportation system and about NASA," he told UPI at the time. "I came into this with seven years experience on the space subcommittee so I didn't start on ground zero.
"But I never had an opportunity to have two months of intense preparation getting to know about the system as well as getting to know the professionals who run the system, both in space and on the ground."
He later wrote a book about his experience titled Mission: An American Congressman's Voyage to Space.