Senate GOP leaders have agreed to roll back $350 billion in tax relief in response to a procedural ambush by deficit hawks led by Sen. Bob Corker Robert (Bob) Phillips CorkerHas Congress captured Russia policy? Tennessee primary battle turns nasty for Republicans Cheney clashes with Trump MORE (R-Tenn.) that nearly killed the GOP tax-reform bill.

Senate Republican Whip John Cornyn John CornynTumultuous court battle upends fight for Senate Texas Democrats roll out first wave of planned digital ads as Election Day nears Calls grow for Biden to expand election map in final sprint MORE (Texas) told reporters after a round of intense discussions on the floor, “We have an alternative, frankly, tax increase we don’t want to do to try to address Sen. Corker’s concerns.”

Cornyn said the details of the proposal are being worked out.

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Corker had insisted on a trigger proposal that would have rolled back tax relief in case economic projections fell short of expectations.

But the Senate parliamentarian ruled Thursday afternoon that the trigger would not pass procedural muster.

“It doesn’t look like the trigger’s going to work, according to the parliamentarian,” Cornyn said.

Sen. David Perdue (R-Ga.) said the estimated reduction in tax relief would be $350 billion over a decade.

Sen. John Hoeven John Henry HoevenDavis: The Hall of Shame for GOP senators who remain silent on Donald Trump Bottom line Bipartisan senators seek funding for pork producers forced to euthanize livestock MORE (R-N.D.) said adding an alternative minimum tax for wealthy individuals and large corporations in the later years of the tax bill is under discussion.

Hoeven said "my understanding is that it would likely apply to C-corps and higher-income individuals."

Corker had joined with two other deficit hawks, Sen. Ron Johnson Ronald (Ron) Harold JohnsonThis week: Supreme Court fight over Ginsburg's seat upends Congress's agenda GOP set to release controversial Biden report Democrats fear Russia interference could spoil bid to retake Senate MORE (R-Wis.) and Sen. Jeff Flake Jeffrey (Jeff) Lane FlakeJeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Republican former Michigan governor says he's voting for Biden Maybe they just don't like cowboys: The president is successful, some just don't like his style MORE (R-Ariz.), and threatened to vote for a motion to recommit the tax bill back to the Finance Committee.

That move would have put the legislation in limbo for the foreseeable future and scuttled an all-night voting session on tax relief.

Conservatives pounced on Corker on Thursday night, accusing him of breaking his agreement with Sen. Pat Toomey Patrick (Pat) Joseph ToomeyAppeals court rules NSA's bulk phone data collection illegal Dunford withdraws from consideration to chair coronavirus oversight panel GOP senators push for quick, partial reopening of economy MORE (R-Pa.) to set the total size of the tax package at $1.5 trillion.

“If tax bill is less than $1.5 trillion, then Corker broke his deal with Toomey. He is a dishonest broker,” tweeted Club for Growth, a group that advocates for lower taxes and less government.

Senators have to figure out a way to ensure that their proposal meets the complex budgetary rules that GOP leaders are using to pass the tax bill with only 51 votes instead of the 60 usually required to pass major legislation.

An analysis released Thursday by the Joint Committee on Taxation found the bill's macroeconomic effects would reduce the deficit by $408 billion over 10 years, but still cost about $1 trillion during that time. The committee had earlier estimated that the plan would lose $1.4 trillion in federal revenue before accounting for economic growth.

Corker and Flake originally wanted to establish a trigger that would have reduced the size of the tax package by $350 billion in case it failed to produce as much economic stimulus as predicted.

But when the parliamentarian nixed that proposal, they insisted on reducing the overall size of the package, regardless of whether it revs up the economy enough to substantially increase further federal revenues.

“My understanding is that the parliamentarian has ruled against it so they’re going to put it in,” Corker told reporters.

The new proposal under discussion would reduce the total size of the tax package by $350 billion starting in year six or year seven of the next decade.

Cornyn told reporters that the size of the reduction could be even larger.

Republican leaders appeared extremely frustrated with Corker, Flake and Johnson during their intense discussions Thursday night while the fate of the bill teetered in the balance.

McConnell’s face grew flushed as he huddled with Corker and Toomey, one of the main architects of the tax reform bill, while GOP colleagues crowded around them to listen in.

Johnson said he joined Corker’s rebellion so he could win an assurance from GOP leaders about getting a vote on setting the corporate tax rate above the 20-percent level favored by President Trump.

– Jordain Carney and Nathaniel Weixel contributed

Updated: 6:38 p.m.