Members of the Senate’s Gang of Six began on Wednesday to sell the group’s bipartisan deficit-reduction proposal to House Democrats as the likelihood grew that lawmakers would need to approve a short-term increase to the $14.3 trillion debt limit.

The pressure of the ticking clock weighed on officials both in the White House and at the Capitol, who cheered the new Senate proposal but acknowledged there was not enough time to enact it before the Aug. 2 deadline to avoid an unprecedented default by the U.S. government.

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With 12 days remaining, President Obama summoned congressional leaders of each party to the White House for separate meetings, and his chief spokesman said for the first time that Obama would sign a “very short-term” extension of the debt limit if Congress agreed to a long-term deficit-reduction deal. The president had previously said he would veto any stopgap measure.

“The president does not support a short-term extension of the debt limit, period,” White House press secretary Jay Carney said. “The only exception to that is in the event that both sides reach a deal on a long-term extension of the debt limit plus significant deficit reduction, and we needed a very short-term extension, like a few days, to allow a bit of extra time for a bill to work its way through the legislative process.”

The Gang of Six proposal, which would slash the deficit by $3.7 trillion over a decade, appeared to be the most promising avenue for a comprehensive deal after it won bipartisan support in the Senate upon its release Tuesday. The Gang’s Democratic members, Sens. Dick Durbin Richard (Dick) Joseph DurbinSupreme Court nominee gives no clues in GOP meeting Warren won't meet with Barrett, calling Trump's nomination an 'illegitimate power grab' Conservative group unveils ad accusing liberals of attacking Barrett's faith MORE (Ill.), Kent Conrad (N.D.) and Mark Warner Mark Robert WarnerTrump official releases unverified Russian intel on Clinton previously rejected by Senate panel FBI director casts doubt on concerns over mail-in voting fraud Democrats call for declassifying election threats after briefing by Trump officials MORE (Va.), held briefings with several dozen House Democrats, who received the proposal warmly, lawmakers said.

“The sentiment was a lot of hope and a lot of encouragement,” said Rep. Ron Kind Ronald (Ron) James KindWisconsin Rep. Ron Kind wins primary Democrats exit briefing saying they fear elections under foreign threat Bottom line MORE (D-Wis.), the vice chairman of the New Democrat Coalition, a business-friendly bloc that heard a briefing from Warner. “It’s the one glimmer of hope that we have around here.”

That optimism was tempered, however, by the dwindling time and the lack of detail and legislative language in the Gang of Six proposal.

“Given the fact the clock is ticking … there’s not going to be an opportunity to take any specific action on this proposal between now and August 2,” Rep. Chris Van Hollen (Md.), senior Democrat on the House Budget Committee, told The Hill. “We’ll have to look at our options.”

Van Hollen organized a separate briefing for a group of Democrats with House Minority Whip Steny Hoyer (Md.).

House Republicans have yet to embrace the Gang proposal, but critically, their leaders have not rejected it. Speaker John Boehner John Andrew BoehnerLongtime House parliamentarian to step down Five things we learned from this year's primaries Bad blood between Pelosi, Meadows complicates coronavirus talks MORE (R-Ohio) and Majority Leader Eric Cantor Eric Ivan CantorThe Hill's Campaign Report: Florida hangs in the balance Eric Cantor teams up with former rival Dave Brat in supporting GOP candidate in former district Bottom line MORE (R-Va.) gave the plan mixed reviews on Tuesday night. Top committee chairmen Paul Ryan Paul Davis RyanBiden's debate game plan? Keep cool and win Trump, Biden have one debate goal: Don't lose RNC chair on election: We are on track to win the White House MORE (R-Wis.) and Dave Camp (R-Mich.) largely echoed the leadership, applauding some of the plan’s budget reforms while criticizing elements of its tax and entitlement proposals.

“Unfortunately, it increases revenues while failing to seriously address exploding federal spending on healthcare, which is the primary driver of our debt,” Ryan, the Budget Committee chief, said in a statement. “There are also serious concerns that the proposal’s substance on spending falls far short of what is needed to achieve the savings it claims.”

The Gang of Six proposal would raise more than $1 trillion in revenues, but members say the Congressional Budget Office would score it as a net tax cut because it lowers personal income rates and abolishes the Alternative Minimum Tax.

Yet because the plan ends the George W. Bush-era tax cuts and raises revenue through other means, House Republicans who have sworn off any tax increases were hesitant to embrace it.

“I think it’s a positive development that there’s some bipartisan approach to addressing the debt limit, and I like the fact that there’s lower rates,” Camp told The Hill in a brief interview, “but I’m concerned about the revenue that’s raised that isn’t because the economy’s growing but because there’s a tax increase. I’m concerned about that.”

House Republicans have not been fully briefed on the plan, and the conference chairman, Rep. Jeb Hensarling (R-Texas), said he was not aware of plans to do so. The Democratic members of the Gang of Six said they had asked their GOP partners in the group, Sens. Saxby Chambliss Clarence (Saxby) Saxby ChamblissLobbying world GOP lobbyist tapped for White House legislative affairs The Hill's Morning Report - Gillibrand drops out as number of debaters shrinks MORE (Ga.), Mike Crapo Michael (Mike) Dean CrapoSupreme Court nominee gives no clues in GOP meeting Barrett to sit with McConnell and other GOP senators in back-to-back meetings Bottom line MORE (Idaho) and Tom Coburn Thomas (Tom) Allen CoburnCOVID response shows a way forward on private gun sale checks Inspector general independence must be a bipartisan priority in 2020 Congress must protect federal watchdogs MORE (Okla.), to reach out to House Republicans.

The Senate proposal drew criticism — some of it predicted — from lawmakers and interest groups on the left and right. Sen. Bernie Sanders Bernie SandersTrump, Biden clash over health care as debate begins Biden calls Trump a 'liar' and a 'clown' at first debate Biden mocks Trump campaign debate claims: 'I've got my earpiece and performance enhancers ready' MORE (I-Vt.) deemed the plan a “victory” for Republicans, and Rep. Raul Grijalva (D-Ariz.), co-chairman of the Congressional Progressive Caucus, called it “terrible” for its cuts to entitlement programs. Labor leaders also assailed the proposal for achieving budget austerity on the backs of the working class, the elderly and the poor.

On the right, the chairman of the conservative Republican Study Committee, Rep. Jim Jordan (Ohio), came out against the gang’s plan.

Yet despite the criticism, it appeared to have life in the House, due in large measure to the persistent opposition from conservatives to the primary backup proposal for avoiding a default: Senate GOP Leader Mitch McConnell Addison (Mitch) Mitchell McConnellIn rare move, Schumer forces vote to consider health care bill amid Supreme Court tensions COVID-19 talks hit crucial stretch Supreme Court nominee gives no clues in GOP meeting MORE’s (Ky.) plan to raise the debt ceiling while granting Obama more power to approve future increases in federal borrowing.

GOP lawmakers and aides said distaste with the McConnell proposals was as strong as ever, giving new momentum to the pursuit of a “grand bargain.”

The immediate goal, some lawmakers said, was to broaden the support for the Gang of Six plan and to figure out how to buy enough time to get it passed.

“This is the first dance of the Senate with the House,” Kind told The Hill. “We just need to invite more colleagues to the party.”