Washington (CNN) Republican leaders are trying to stem an issue that's causing division within their own party and threatening to stall movement in the tax reform push they're so desperately trying to win.

On the chopping block is the State and Local Tax -- or SALT -- deduction , a popular tax break that affects nearly one-third of filers and lets them deduct levies like state income taxes and property taxes. It's been in place since the birth of the federal income tax in 1913.

Supporters of the deduction argue it helps taxpayers avoid double taxation in states with high taxes like New York, New Jersey, California and Illinois. While those states are largely run by Democrats, there are several Republican members of the House with districts in those states, and they are staunchly defending the deduction, arguing its loss would impact constituents who've been relying on it for their whole tax-paying lives.

"We're fighting it with everything we know how," said Republican Rep. Tom MacArthur of New Jersey.

Intra-party fight

Critics say the deduction unfairly subsidizes high-tax states and only encourages state governments to levy higher taxes on residents, knowing they can write them off on their federal returns. House Speaker Paul Ryan said Thursday it creates a situation in which "states that actually got their act together pay for states that didn't."

After giving a speech at the Heritage Foundation, Ryan acknowledged that SALT has become a speed bump in the tax reform push, describing the issue as "the thing we have to get over." But he argued it's a sacrifice for the greater good.

"The general interest is going to have to trump over the special interest," he said.

That's in part because getting rid of it could bring in an estimated $1.3 trillion over the next 10 years -- a huge source of revenue that the tax-writing committees in Congress are counting on to help pay for tax reform.

Politically, however, it's tricky. Republicans can only afford to lose about 22 votes on tax reform, and there are more than 30 Republicans in the House who represent districts in these high-tax states.

Rep. Leonard Lance called his home state of New Jersey a "sending state," saying it gets back disproportionately less money from Washington than it sends in taxes. "I do not want to exacerbate that situation," he said, adding that he wants to see the deduction remain as is with no changes.

Lance helped lead other New Jersey Republicans -- like Rep. Frank LoBiondo, Rep. Rodney Frelinghuysen and Rep. Chris Smith -- to team up with Democrats in a letter to Treasury Secretary Steve Mnuchin this summer, urging him to keep the deduction in place.

New York Republicans did the same with six members, including Rep. Peter King, Rep. Dan Donovan and Rep. Lee Zeldin, signing on.

Potential compromise?

Rep. Kevin Brady, who chairs the House Ways and Means Committee (which is writing the tax bill), has been working behind closed doors with these Republicans and hearing out their concerns. He met with members Thursday afternoon in Speaker Ryan's office, and last week they dined at Ruth's Chris steakhouse in Washington.

"We'll continue those discussions until the tax reform bill is introduced," Brady told reporters this week. "We want their support."

Republicans emerged from Thursday's meeting saying there was "progress" but gave no specifics on the potential deals that were discussed. Various proposals have been floated publicly in recent weeks, including the idea of capping the income level at which a taxpayer can deduct. That would prevent wealthier taxpayers from using the deduction.

"There's more than one way to skin this cat. I'm not fixated on one solution," said MacArthur, who's offered up his own proposal that would let New Jersey taxpayers, who have the highest property taxes in the country, continue to deduct their property taxes. "I think we're going to solve this issue."

Rep. Tom Reed, a member of the House Ways and Means Committee and a Pennsylvania Republican, said "I think we're moving closer, but we have a lot more work to do."

Yet several sources involved in the process say significant changes, at this point, are unlikely and certainly not desired by the so called Big Six, the group of tax negotiators in Congress and the Trump administration that's been working all year on the plan.

Senate Republicans, unburdened by delegations of Republicans from the states that would be the hardest hit, are on board with the deduction's repeal, adding a new lever of pressure as the process kicks into gear in the weeks ahead.

Focusing on the 'big picture'

For now, even as alternatives are given a hearing and smaller scale tweaks are considered, the broader goal on the part of leadership is to stress the end game, the sources said. In other words, they want members to focus on the overall net benefit that they believe will leave states in a better place, even without the deduction.

And that's exactly what Ryan did on Thursday. He made no mention of a potential deal but said members need to focus on "the big picture." Middle class taxpayers, he argued, will benefit "no matter what state you come from" because of other changes to the tax system, like a doubling of the standard deduction and the lowering of individual tax rates, as proposed in the plan.

It's a tough sell, especially as some of the most important numbers in the plan remain shrouded in secrecy. Republican Rep. Dave Brat, who represents a district in Virginia where more than four in 10 taxpayers take advantage of the deduction, is in favor of getting rid of the tax break because it would help keep individual rates down.

But he said it's hard to communicate that to constituents when so little is known about the individual tax rate part of the plan.

"If you had all the details, then people could get out the calculator and get out a little app and see if they're better off or worse off," he said. "To do that, you need to have your brackets and everything finalized. Everyone right now is just looking at 'hey, I might lose a little on this thing,' and they don't know how much they win will be on the other thing."

Rep. Patrick McHenry, the chief deputy whip in the House, said leadership can resolve the problem "economically" for members, but in a stark acknowledgment of the unfavorable optics that concern many Republicans, he wondered whether it can be squared away "politically" in these districts.

"And that takes members' will to follow through on that," the North Carolina Republican said at an event Wednesday sponsored by the Financial Services Roundtable. "To say the net benefit is better than any one provision."