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Washington (CNN) -- With a possible government shutdown in five days, House Speaker John Boehner on Thursday signaled a shift in Republican tactics with the focus now on negotiating a compromise on raising the federal debt ceiling next month instead of a short-term spending plan now.

He and other GOP leaders called on President Barack Obama to negotiate an agreement with them to increase how much the government can borrow to pay its bills in exchange for concessions they seek such as delaying the implementation of Obamacare.

However, Obama later made clear that he rejected any political bartering on the debt ceiling.

"I will not negotiate on anything when it comes to the full faith and credit of the United States of America," he said.

It is one of two potential showdowns embroiling Congress as deadlines loom to fund the government before the end of the current fiscal year on Monday, as well as increasing the debt ceiling by October 17.

The Democratic-led Senate is considering a spending plan sent over by the Republican-led House that makes continued government funding contingent on eliminating all money for Obamacare.

Senate Majority Leader Harry Reid says his caucus will restore the funding for the Affordable Care Act, the signature legislation of Obama's first term that was passed by Democrats and upheld by the Supreme Court.

The spending measure would then go back to the House with time running out to act before a shutdown begins on Tuesday.

Fed up over Beltway shutdown battle

GOP battle line moving to debt ceiling

At a news conference after meeting with his GOP caucus, Boehner indicated the House might further revise the Senate's version of the spending plan, but he also told reporters that he didn't expect a government shutdown to happen.

Instead, he and other House GOP leaders focused on the debt ceiling, with Majority Leader Eric Cantor of Virginia citing a Bloomberg poll Thursday that showed a strong majority support a compromise on the debt ceiling instead of simply increasing it without spending cuts or other steps.

"They don't want a government shutdown, but they are tired of the debt and they want to make sure that this president sits down and negotiates with us a resolution to this problem," Cantor said. "So we call on the president now to sit down with us, Harry Reid to sit down with us, and let's solve the problem. This plan of more debt and no reform is absolutely unacceptable."

Cantor confirmed that a House proposal to raise the debt ceiling would include a one-year delay in the full implementation of Obamacare, as well as other provisions from measures passed by House Republicans but ignored by the Senate such as building the Keystone oil pipline from Canada.

To Obama, such an offer amounts to trying to "blackmail a president into giving them some concessions on issues that have nothing to do with the budget," he said in a speech in Maryland on Obamacare.

White House spokesman Jay Carney told reporters a one-year delay in implementing Obamacare's individual mandate for people to obtain health insurance would undermine a key provision of the program that prohibits the denial of coverage due to pre-existing conditions.

"The fact is you have to make the system work," Carney said, adding people with pre-existing conditions won't be denied insurance under Obamacare "because of the expansion of the number of people who will be covered and participate in these marketplaces provided by the Affordable Care Act through the individual mandate."

CNN Chief National Correspondent John King said Thursday that focusing on the debt ceiling was where House Republicans "wanted to wage this fight all along."

"They didn't want to get bogged down in the government shutdown fight, but a conservative revolt within the House Republican ranks forced them to get there," King said.

How this affects you

Conservative backlash

However, some conservative House Republicans emerging from a meeting Thursday with Boehner said they won't simply accept the Senate's version of the spending plan needed to avoid a shutdown. They also expressed reservations about the proposal outlined by Boehner and Cantor for raising the debt ceiling.

Rep. Mo Brooks of Alabama said the GOP leadership's plan on the debt ceiling would extend it until after next year's congressional election and include provisions sought by Republicans, but he complained it didn't cut spending enough to satisfy him. Brooks also said the debt ceiling issue should be part of the election debate next year.

Treasury Secretary Jack Lew said this week the debt ceiling must be raised by October 17, or the government may not have enough money to pay all of its bills.

Analysts warn of severe economic impact in that case. A similar bout of congressional brinksmanship over the debt ceiling in 2011 led to the first-ever downgrade of the U.S. credit rating.

Obama cited that potential harm to a still sluggish recovery on Thursday, saying the world looked to America for economic leadership and stability.

"You don't mess with that," he said to cheers in Maryland.

However, King said the president previously negotiated on the need to increase how much the federal government can borrow to meet its obligations, such as in the 2011 showdown that led to deep spending cuts sought by Republicans.

"The Republicans think that is safer ground politically. The question is, can they actually get anything from the president?" King said.

Noting Obama has previously accused Republicans of having an unreasonable "my way or the highway" stance, he said the GOP can now accuse the president of doing the same thing by refusing to negotiate on the debt ceiling.

Senate spending plan

In the Senate, tea party conservatives led by GOP Sen. Ted Cruz of Texas reversed themselves on Wednesday by joining colleagues from both parties in a 100-0 procedural vote to take up the House spending plan expected to be revised to fund Obamacare.

Cruz supported the move after he spent more than 21 straight hours railing against any government money for the health care reforms.

The rare unanimous vote on a procedural step signaled that Senate Democrats would be able to amend the plan to restore the Obamacare funding.

On Thursday, Reid made clear that if the House then made further changes to the revised spending proposal sent back by the Senate, it would cause at least a short-term government shutdown because of the time it would take the Senate to reconsider the measure.

Cruz had led a group of tea party conservatives in trying to block Senate consideration of the spending legislation.

However, he came under strong criticism from fellow Republicans including Senate GOP leader Mitch McConnell of Kentucky and other influential veterans such as Sens. John McCain of Arizona and Bob Corker of Tennessee for that strategy, which called for GOP senators to filibuster the House measure that -- in its original form -- would defund programs under the Affordable Care Act.

The Obamacare question

Corker told CNN on Wednesday that a better idea would be to get the bill back to the House as soon as possible so the Republican majority there can offer a compromise.

"House members are already talking about how they might respond if the defunding component ends up being stripped out," Corker said, adding he hoped that the Senate would "give the House some time to respond in a thoughtful way."

With Obamacare markets for the uninsured set to open on October 1, which also begins the new fiscal year, GOP opponents consider this their last best chance to undermine or amend the health care reforms.

What happens in a government shutdown

GOP infighting over how best to prevent a government shutdown while defunding Obamacare escalated this week as McConnell publicly dismissed Cruz's more confrontational strategy.

Cruz's GOP critics believe his strategy is politically suicidal, arguing there is no way to stop Obamacare as long as Democrats maintain control of the Senate and Obama is in the White House.

They believe that trying to do so by forcing a shutdown -- or preventing an increase in the debt ceiling next month -- will backfire by harming the economy and damaging the Republican brand.

5 strange things about debt ceiling politics

CNN's Deirdre Walsh and Bryan Koenig contributed to this report.