Washington (CNN) -- Congress will agree to extend the payroll tax cut before it expires at the end of the year, two leading conservative Senate Republicans said Sunday.
The comments demonstrated the GOP shift on a proposal originally opposed by Republican leaders, even as each side continues trying to persuade the other of the best legislative approach to take.
Senate Minority Leader Mitch McConnell told "Fox News Sunday" that bipartisan support in both chambers would coalesce around a House Republican plan that links extending both the lower payroll tax rates and unemployment benefits to easing the path for approval of the Keystone XL oil pipeline from Canada.
President Barack Obama and Senate Democratic leader Harry Reid have rejected the House plan, but McConnell, R-Kentucky, said enough Senate Democrats support it to get it passed if a vote is held.
Asked if Americans will wake up January 1 to a tax increase because Congress failed to act on the issue, McConnell responded: "That isn't going to happen, and obviously we'll reach an agreement." Later in the interview, McConnell declared that extending the payroll tax cut "has bipartisan support."
Another conservative Republican, Sen. Lindsey Graham of South Carolina, agreed that a compromise would emerge but said he doubted it would include the oil pipeline provision.
"At the end of the day the payroll tax will get extended as it is now," Graham said on NBC's "Meet the Press," adding: "The pipeline's probably not going to sell."
Democrats led by Obama have been pushing for the payroll tax cut extension, with Republicans initially opposing the plan but now signing on. However, the parties are to loggerheads over how to pay for it, as well as the political optics over who should get credit as an election year approaches.
"It's the highest priority of the president and the Democrats in Congress," liberal Sen. Dick Durbin of Illinois told NBC.
The House GOP proposal would both change the process and shorten the time frame for the government to approve the proposed pipeline from Canada's oil sands production in northern Alberta to Texas.
The State Department recently said its decision would be delayed until 2013 to examine environmental issues raised by critics, a move Republicans labeled as political to put off the issue until after next year's presidential election.
Obama said last week that he will reject any effort to link the oil pipeline to extending the payroll tax cut, which will maintain $1,000 in tax savings for American workers.
"Now is not the time to be debating unrelated measures like an oil pipeline," Reid said in a statement Friday. "If the House sends us their bill with Keystone in it, they are just wasting valuable time because it will not pass the Senate."
The impasse sets up another countdown clock on congressional action, this time before the end-of-year holiday recess scheduled to begin at the end of next week.
The House GOP measure extends the payroll tax cut for one year and renews aid for the unemployed, while cutting back the maximum length of jobless benefits from the current 99 weeks to 59.
The bill also allows states more flexibility in distributing unemployment assistance, permitting states to require those applying to submit to drug tests or show they are pursuing a high school degree if they don't have one. The bill would also avoid a scheduled cut in pay for Medicare physicians for two years, a provision known as the "doc fix."
As a way to garner support from reluctant conservative Republicans, who voiced concerns about the impact of the payroll tax cut on Social Security, GOP leaders insisted on keeping the provision aimed at moving toward approving the pipeline project within 60 days.
To pay for the bill, GOP leaders use a series of spending cuts, including freezing pay for federal employees and members of Congress, eliminating a child tax credit for those in the U.S. illegally, and increasing Medicare premiums for those who earn more than $80,000 annually.
The pipeline issue is just one part of the problem for Senate Democrats who disagree with "central elements" of the GOP bill, including the length of the unemployment insurance extension and the proposed cost offsets, according to one Democratic aide.
House Democratic Leader Nancy Pelosi of California argued Friday that the pipeline provision is "a poison pill designed to sink the payroll tax cut."
The House is expected to vote on the Republican bill Tuesday. In the meantime, Senate Democrats have begun working on a new bill of their own after Republicans blocked progress on previous proposals that included a surtax on income over $1 million.
"We'll find a way to pay for it in a bipartisan fashion," Graham said on NBC, adding that the "idea of taxing one group to pay for a tax cut for another is not going to sell."
Durbin complained that Republicans "have consistently said they will refuse to increase the taxes on the wealthiest people in America one penny, if that's what it takes to make sure that working families get a payroll tax cut."
CNN's Ted Barrett and Deirdre Walsh contributed to this report.