The proposal, unveiled in April as part of Mr. Obama’s budget, is far from dead. On Tuesday, a committee in the Democrat-controlled Senate is scheduled to work on its version of the bill, one that makes no mention of capturing asteroids but gives leeway to NASA to do whatever it thinks best for getting to Mars. On the same day, experts will convene at NASA headquarters in Washington to review work on the asteroid mission so far.

As yet, those experts have not pinpointed an asteroid to kidnap, but the idea is this: First, build a robotic spacecraft with a novel inflatable cone-shaped structure that could envelop the asteroid (which will be tricky to catch, because it will probably be spinning). Next, meet the space rock as it swings by the vicinity of the Earth and the Moon. Then, after essentially wrapping the asteroid in a bag (no lassos are actually involved), the spacecraft would lug it into orbit above the Moon, a slow do-si-do of mechanics that could take a few years.

“Over all, I think this is a very doable mission,” said Brian Muirhead, the chief engineer at the Jet Propulsion Laboratory.

After that, astronauts would travel aboard a giant new rocket that NASA is designing, to meet the asteroid for a closer look. Their trip would give NASA the opportunity to test its deep-space spacecraft, the Orion capsule, as well as its procedures for helping astronauts work with asteroids, which have almost no gravity.

Asteroids have been having their moment in the news, in part because of the terrifying asteroid explosion over Russia in February, which injured about 1,500 people. Last month, NASA announced an Asteroid Grand Challenge, inviting people and organizations to collaborate in finding asteroids that threaten Earth and proposing solutions. On Friday, the agency said it had received more than 400 responses to the challenge and suggestions to help with the asteroid capture mission.

Separately, at least two private companies have announced intentions to mine asteroids for rare metals, arguing that supplies on Earth are dwindling.

There is near unanimity in Congress and NASA that the ultimate goal is to send people to Mars, but the logistical challenges and costs are too big to conquer right away. NASA officials depict the asteroid capture plan as an elegant interim step, one that would send humans deeper into space than before and break new ground in rocket technology.