Some Republican lawmakers, raising new alarms over the effects of automatic spending cuts on defense, are opening the door to new revenue with only days to go until the sequester hits, in contrast with widespread opposition to additional taxes among most in the GOP.

Sen. Lindsey Graham, R-S.C., said he could support another $600 billion in revenue if it were part of a larger deficit reduction deal including an overhaul of Medicare and Social Security.

And in a trip to Newport News, Va., with President Barack Obama on Tuesday, Virginia Republican Rep. Scott Rigell, who represents the district that includes a large military presence around Newport News and Norfolk, said he favored raising additional revenue as part of a tax overhaul.

“Our country has a spending problem,” he said. “We need to grow our economy and raise revenues that way. I also believe that revenue has to come up a bit, first by growing the economy but also by tax reform, which also includes eliminating lobbyist-inspired, lobbyist-written loopholes. I am in favor of that.”

House and Senate GOP leaders have ruled out additional revenue after the fiscal agreement at the start of this year, which is projected to raise more than $600 billion in new taxes over a decade. They reiterated that stance on Tuesday as it became clear that Friday’s onset of the automatic, across-the-board spending cuts will start to hit government programs and agency operations.

“There have been four years of spending increases, and now the president says we can’t have any progress on the sequester unless he gets a second tax hike in eight weeks,” House Majority Leader Eric Cantor of Virginia said at a news conference with other House GOP leaders.

The debate over the sequester has come down to a battle over the new tax revenue the president says should be part of what he calls a “balanced” approach to replace the automatic reductions and Republican calls for newly designed reductions in domestic spending. Rigell and Graham are on the Armed Services panels in their respective chambers, and both have spoken out against the effects of the sequester on defense.

Graham said Tuesday he would back an additional $600 billion in revenue if that were combined with changes to reduce the growth of entitlement programs, including raising the Medicare eligibility age, means testing and a new inflation index for entitlement programs. But he stressed the big deal he is seeking is aimed primarily at stabilizing the growing, more than $16 trillion in national debt, not replacing the $1.2 trillion sequester.

“I’m not going to put any revenue on the table to fix sequestration,” he said. “The next time I would vote to increase revenue would be when we did entitlement reform.”

“Do like what Simpson-Bowles did,” Graham said, referring to the commission that called for a range of spending cuts and tax provisions to reduce the deficit over time. “Put some money on debt retirement, some money on rate reduction and reform entitlements in a way that you’re not going to privatize Social Security, you’re not going to create a voucher program for Medicare. I’m talking about means testing, CPI adjustments and age adjustments.”

Rigell argued that more revenue is needed and said the GOP should not rule out any proposal just because it may include some revenue. But he framed additional revenue as part of a larger agreement.

Rigell noted House Republicans passed two sequester replacement bills last year and called on Obama to offer a detailed plan for stopping the sequester. Both would have replaced the automatic cuts with reductions in domestic programs. The second passed late in the year with a 215-209 vote, and it is unclear whether the House leadership could muster the votes to pass that bill in the 113th Congress.

The Senate plans to vote on Democratic and Republican legislation to replace the first year of sequestration Thursday. Although the GOP has not settled on a plan yet, Sen. Michael D. Crapo, R-Idaho, said on the way into Senate Minority Leader Mitch McConnell’s office Tuesday that he expects the plan would give Obama flexibility to reallocate the spending cuts.

Graham is among at least a handful of GOP senators who are skeptical of giving the administration that authority.

Graham said his support for additional revenue is not at odds with McConnell’s opposition to additional taxes. “No, I agree with Sen. McConnell,” Graham said. “We don’t need any more tax revenue to fix sequestration. I believe Sen. McConnell would do more revenue for entitlement reform.”

McConnell spokesman Mike Brumas dismissed the need for new revenue. “The leader has already addressed this issue in a dozen speeches this year — the tax debate is over,” he said.