Meanwhile, President Barack Obama and Democrats want a "clean" increase and insist they won't negotiate.
Wait ... is this the same fight they're having over a potential government shutdown? No. But it's easy to confuse the two because they're happening at the same time.
The government will shut down if Congress can't agree to fund the government before the fiscal year starts on Tuesday. And Republicans are making similar demands to what they are making over the debt ceiling.
But here's the key difference: Shutting the government down would be aggravating and a waste of time -- and could reduce economic growth if it goes on too long. But that's nothing compared to the risks posed if Congress doesn't raise the debt ceiling soon.
So what happens if Congress doesn't raise it in time? No one knows for sure because that's never happened before. But the going assumption is that no good will come of it.
Treasury would not be allowed to borrow money. And that's a problem since the government borrows to make up the difference between what it spends and what it takes in.
Practically that means at some point this fall -- the updated estimate is late October to early November -- the Treasury will no longer be able to pay all the country's bills, benefits and other obligations in full and on time.
What then? That's a good question with no clear answer.
The Treasury may try to pay some bills and delay others, or delay all bills due on a given day until it has enough revenue in hand to pay all of them.
Most experts think the Treasury would do all it could to prioritize interest payments on the debt, lest the United States default on its bonds, which would likely send markets plunging and interest rates soaring.
But it's not clear how investors will respond if Treasury makes interest payments but ends up delaying payments to government contractors, federal workers, taxpayers due refunds, veterans, seniors and anyone else to whom the federal government has a legal obligation.
Economically it could be disastrous if the standoff lasts for more than a couple of days.
"Federal employees, contractors, program beneficiaries, businesses and state and local governments would find themselves suddenly short of expected cash, causing a ripple effect through the economy," Donald Marron, a former Congressional Budget Office director, told lawmakers.
To say nothing of the fact that the United States' reputation as "money-good" would be damaged.