“It’s not going to happen overnight — we’ve been debating this for 30 years,” said Mr. Lott, mentioning past efforts like the failure to pass complicated cap-and-trade legislation in 2009 and the lack of support for the Kyoto Protocol climate agreement. But now, he said, “I really do think the tide is turning.”

Mr. Lott pointed to bipartisan support in the current Congress for renewing a tax break for technologies to capture and store carbon dioxide, and cited results of a poll commissioned by the group that showed wide support for the principles of taxing carbon and returning the money to taxpayers. A 2016 study by the Yale Program on Climate Change Communication also found that most Americans would back such a program.

Giving the money back to consumers could sweeten the deal for taxpayers, but other parts of the plan are intended to foster support from other quarters. Climate change regulations on businesses would be rolled back — although the Trump administration has already moved to rescind a number of existing rules. Fossil fuel companies, meanwhile, would be protected from lawsuits intended to hold them accountable for the damage caused by climate change.

This element of the deal has angered some climate change activists. “We categorically oppose any deal that shoves trillions in costs down the throats of innocent taxpayers and lets the companies skip away to profit and deceive another day,” said Lee Wasserman, director of the Rockefeller Family Fund, which has encouraged investigations and lawsuits against the fossil fuel industry.

Still, the group has drawn bipartisan support. Mr. Lott is a co-chairman, with John B. Breaux, a former Democratic senator from Louisiana. Senior advisers to the group include Mark McKinnon, who was an aide to President George W. Bush, and Joe Lockhart, who worked with President Bill Clinton. Ben S. Bernanke, Ms. Yellen’s predecessor, is another founding member.

The plan was first put forth in February 2017 by a related group called the Climate Leadership Council, with the backing of a team of Republican lions like the former secretaries of state James A. Baker III and George P. Shultz.