Updated: 7:27 p.m.

Senators in both parties on Monday appeared to be grudgingly warming to a $2.1 trillion deficit reduction package agreed to by Congressional leaders and President Barack Obama.

Though Senate Majority Leader Harry Reid (D-Nev.) said Monday he is “hopeful” the bill to raise the debt ceiling will pass as soon as Tuesday evening, a senior Democratic aide later a vote would not be held until Tuesday morning. The House passed the bill, 269-161, Monday evening.

Emerging from a caucus luncheon with his Members and Vice President Joseph Biden, Reid said some of his rank and file were enthusiastic and some were not, but he hopes the legislation will clear his chamber with help from Republicans.

With the Treasury Department needing an extension of the nation’s borrowing capacity by Aug. 2, the Senate’s Tuesday vote will come perilously close to the deadline. However, rank-and-file lawmakers on both sides of the aisle told reporters they expect strong support for the package as an alternative to a potentially catastrophic default.

However, some Senators still expressed concerns about the details of the agreement. Liberals worried the cuts were too deep and lamented the lack of any revenues, while conservatives wanted even deeper cuts and, at the same time, are nervous about a trigger that could slash defense spending.

“I feel like I’ve just met with my doctor and he’s told me what I need to do is diet and exercise,” said Sen. Barbara Mikulski (D-Md.), who added that she is leaning toward voting for it but isn’t happy about doing so.

Sen. Dianne Feinstein (D-Calif.) predicted a majority of Democrats will back the deal because defaulting is simply unacceptable.

Sen. Mark Warner (D-Va.), a co-founder of the Senate’s bipartisan “gang of six,” which proposed a much more ambitious $3.7 trillion plan, said it “doesn’t get us to the core problem” but that he is inclined to support it.

“This is clearly not the larger deal that we need,” agreed fellow group member, Sen. Mike Crapo (R-Idaho), but he also is likely to back it as the first step of many to get the nation’s finances in order.

Republican Conference Chairman Lamar Alexander (Tenn.) touted the package, saying that if they cut an amount equal to the debt limit increase every time the debt ceiling is brought up during the next decade, Congress would be able to balance the budget.

Sen. Tom Harkin (D-Iowa) said he was disappointed at the cuts and the lack of taxes in the deal.

“There will never be a long-term solution until we have revenue,” he said. And he said the deal isn’t going to do anything to put people back to work.

And Sen. Bernie Sanders (I-Vt.) ripped the package because nothing would come from the wealthy and from corporations.

“This deficit reduction package is grotesquely unfair and it is also bad economic policy,” he said.

Despite dissatisfaction, conservative Senators including Jim DeMint (R-S.C.), Rand Paul (R-Ky.) and Mike Lee (R-Utah) said they would not block the Senate from voting on the bill.

Paul and Lee, however, said they wanted time to read and understand the bill — even though they are almost certain to oppose it. That could be the reason for the delayed vote.

Reid faced questions about the composition of a bipartisan 12-Member joint Congressional committee tasked with finding at least $1.2 trillion in additional savings by Thanksgiving. Lawmakers on both sides are fearful that the panel will end in gridlock, which would kick off a series of across-the-board spending cuts to Medicare providers, defense spending and other programs.

Reid told reporters he has two weeks after the bill becomes law to appoint Members and that he talked briefly with Minority Leader Mitch McConnell (R-Ky.) about the matter. The Nevada Democrat emphasized the need to select reasonable Members focused on reaching agreement and not people holding extreme beliefs or “who said gangs are a waste of time” in the first place.

Reid also said he had heard that House Majority Leader Eric Cantor (R-Va.) had indicated a willingness to increase revenue as part of the joint committee, but Cantor’s office denied Republicans are willing to consider tax increases.