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.
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.