While members of the congressional supercommittee are divided on any number of issues, none is more fundamental to the stalemate than taxes — not just whether to raise them, but how.

The tax debate is colored by a long-standing feud between Republicans and Democrats over the fate of the George W. Bush-era tax cuts, which are scheduled to expire at the end of next year.

Republicans have long opposed raising taxes above current levels, which is why they have fought to preserve the Bush tax cuts at a cost to federal coffers of about $4 trillion over the next decade. Democrats also want to preserve the Bush tax cuts, at least for the middle class, at a cost of about $3.2 trillion over the next decade.

The gap between the parties in that debate amounts to the $800 billion that benefits upper-income households. That’s the sum House Speaker John A. Boehner (R-Ohio) offered in new tax revenue in talks with President Obama during this summer’s debt-limit debate.

After those talks collapsed, Congress created the supercommittee to try to bridge the divide — six lawmakers from each party tasked with developing a blueprint to reduce federal borrowing by at least $1.2 trillion over the next decade.

Because the deal produced by the summer debt-limit debate cut government agency budgets by nearly $1 trillion through 2021, Democrats have insisted that the supercommittee now look to new taxes for a significant share of any plan to reduce the deficit. And Democrats continue to view $800 billion as the minimum price for any structural changes to federal health and retirement programs, the biggest drivers of future government borrowing.

For weeks, Republicans refused to budge on taxes. But that changed last week, with an offer from Sen. Patrick J. Toomey (R-Pa.) to raise taxes by about $300 billion over current levels — meaning the amount of taxes collected by the government under a tax code that includes the Bush tax cuts.

About $50 billion would come from using a revised inflation index, which would throw taxpayers more quickly into higher tax brackets. But the rest would come from an overhaul of the tax code that would eliminate lucrative credits and deductions while lowering rates for everyone. Under the Toomey plan, for example, the top rate would fall from 35 percent to 28 percent.

Rewriting the tax code is critical to those Republicans who are willing to raise taxes. The reason is politics. Rewriting the code permits them to argue that while, yes, the total sum of money pouring into the Treasury would increase, people would be subject to lower tax rates and the tax code would be simpler, thus encouraging economic growth.

“The plan that I put on the table is contingent upon pro-growth tax reform,” Toomey, the former president of the anti-tax Club for Growth, said on “Fox News Sunday.” “If there is more revenue, it has to come in the context of pro-growth tax reform, the kind of reform we’re talking about, absolutely guaranteed to create millions of jobs over time and still more revenue.”

Democrats objected to the Toomey plan for two primary reasons: His approach to tax reform is likely to favor the wealthy, because it would not raise rates for investment income. And the plan would generate less than the Democrats’ $800 billion target.

Throughout most of the supercommittee talks, Democrats had agreed to measure any proposed reductions in the deficit based on current tax levels. A senior Democratic aide said they had also explicitly offered to permanently extend all the Bush tax cuts. But the initial Democratic proposals would have clawed back as much as $1.3 trillion through a rewrite of the tax code.

Democrats fundamentally changed that approach in their latest offer, made last Friday. While they dropped their demand for new taxes to just $400 billion, they also retracted their offer to extend the Bush tax cuts. And they insisted that the entire amount be extracted from ending tax breaks for individuals and business, rather than through “pro-growth tax reform.”

On Thursday, Sen. Patty Murray (D-Wash.) insisted that Democrats had matched the Toomey proposal on taxes.

“We have met their offer on revenue, but we have said that it has to be fair to the American people and done in a way that doesn’t put the burden on working families,” she said. “We are waiting for them to accept that.”

But Republicans scoffed at that idea, arguing that the plan left the door open to a tax hike of at least $800 billion on upper-income taxpayers down the road.

“It’s not a $400 billion tax offer,” said a Republican aide close to the process, speaking on the condition of anonymity to discuss the private negotiations. “It’s a $1.2 trillion tax offer that’s worse than the things they have offered before.”