Bill Nelson, chairman of the Senate Science and Space Subcommittee, said the 550-ton space rock could be snatched in 2019.
Local television channel News 13 said that once it had been dragged into a stable orbit, astronauts on board the Orion capsule would mine the rock to learn more about its composition.
It is hoped that four astronauts would nuzzle up next to the rock for spacewalking exploration by 2021, according to a government document.
Donald Yeomans, who leads Nasa's Near Earth Object programme, explained that the asteroid would be captured with the space equivalent of "a baggie with a drawstring".
"You bag it. You attach the solar propulsion module to de-spin it and bring it back to where you want it."
The asteroid would provide scientists with a "unique, meaningful and affordable" destination for the next decade, Senator Nelson was quoted as saying by Florida Today.
Mr Nelson, who represents Florida, told the newspaper that President Barack Obama will put aside $100m (£65m) in planning money for the mission, when the White House unveils its 2014 budget next week.
He said the project would help Nasa defend Earth from a potentially devastating asteroid strike in the future.
It could also help scientists test technologies that could one day be used in a manned mission to Mars.
Last year, the Keck Institute for Space Studies proposed a similar mission for Nasa with a price tag of $2.6bn (£1.7bn). However, the space agency has not yet revealed an estimated cost.
In a separate project, announced in January, US company Deep Space Industries said it planned to send a fleet of spacecraft into the solar system to mine asteroids for metals and minerals.
The first mission could be flown in 2015, with each journey lasting up to six months.