The budget plan could also include a relaxation of the cuts in discretionary spending—the sequester—that are required by the 2011 Budget Control Act.
The ultimate deal may end up looking like the one that House Budget Committee Chairman Paul Ryan, R-Wisc., proposed last week in a Wall Street Journal op-ed: "We could provide relief from the discretionary spending levels in the Budget Control Act in exchange for structural reforms to entitlement programs."
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Ryan—who would be a leader of the conference committee—said what he has in mind "isn't a Grand Bargain," but "modest reforms to entitlement programs and the tax code."
He noted approvingly that President Barack Obama has already called for wealthier retirees to pay higher premiums for their Medicare coverage than they now do.
Under Senate rules, a budget resolution is subject to a special fast-track process. The rules do not allow a member—Sen. Ted Cruz, R-Texas, or any other senator—to use a filibuster to block it. This fast-track process is called "budget reconciliation."
"Probably the most important part of budget reconciliation is that a reconciliation bill can approve policy reforms with only 51 votes in the Senate," said Loren Adler, the research director at the Committee for a Responsible Federal Budget. This fast-track process "could be critical in increasing the chances for tax reform" and reform of entitlement programs to succeed.
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Adler added, "Now that we have lawmakers back to the negotiating table, this is our best opportunity to enact a deal to reform our entitlements and tax code, restore investments and put our debt on a sustainable downward path as a share of the economy."
Of course, there's no guarantee that the conference committee members will be able to agree on a plan.
Taking a pessimistic view was Jason Fichtner, senior research fellow at the Mercatus Center at George Mason University who served as chief economist of the Social Security Administration.
"We've been here before," Fichtner said. "How can we seriously expect any new conference committee to resolve the same difficult decisions that faced the (2011) Super Committee, the National Commission on Fiscal Responsibility and Reform (a.k.a. Simpson-Bowles) and have faced the House and Senate that resulted in this government shutdown in the first place? We can't."
McConnell said Wednesday as the deal was unveiled that he was confident that "we'll be able to announce that we're protecting the government spending reductions that both parties agreed to under the Budget Control Act and that the president signed into law. That's been a top priority for me and for my colleagues on the Republican side of the aisle throughout this debate and it's been worth the effort."
But the conference committee might end up reducing or cancelling some of those Budget Control Act spending reductions.
The goals that conservative Republicans had been seeking since the standoff started last month were not part of the bill Congress passed Wednesday night.
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Sen. Lindsey Graham, R-S.C., said on Twitter: "To say we as Republicans left a lot on the table would be one of the biggest understatements in American political history."
In particular, Republicans did not get:
•Immediate agreed-to entitlement spending reductions or curbs, which was one of House Speaker John Boehner's demands in return for increasing the debt limit.
•A defunding of the Affordable Care Act, or Obamacare, a goal of Rep. Ted Cruz , R-Texas, and his conservative allies, or a delay in Obamacare's individual mandate, the requirement that uninsured people buy health insurance.
•A repeal of or delay in the medical device tax, a part of the Affordable Care Act designed to raise money to help pay the cost of insuring uninsured people. That tax will cost medical device manufacturers and importers $32.5 billion over the next 10 years.
•Ending the policy of the federal government paying a share of health insurance costs for members of Congress and congressional employees, an idea championed by Sen. David Vitter, R-La.
—By Tom Curry, NBC News