After watching the price of oil and the size of their profits plunge, a dozen top executives from some of the nation’s largest oil exploration companies flew to Washington late last winter on an urgent mission: push Congress and the White House to allow unlimited exports of American crude oil.

Now, their long-shot lobbying effort to repeal the 40-year-old export ban has gathered considerable momentum. Approval by the House is expected in the coming weeks, and two Senate committees have already endorsed the idea.

The White House and Senate Democrats may still move to block it, but the fact that the legislation is even moving ahead in an era of extreme gridlock affirms the deep-pocketed oil industry’s durable power in nation’s capital.

“The sooner this happens, the better for us,” said Kenneth P. Cohen, Exxon Mobil’s vice president for public and governmental affairs. “The momentum has picked up. The political calculus right now is very favorable for taking a look and actually doing something about this ban.”