"That’s why the House has acted twice to replace the president’s sequester with common-sense cuts and reforms that protect our national security, and it’s why I’ve been calling on the president for more than a year to press his Democratic-controlled Senate to do the same," BoehnerJohn Andrew BoehnerAre maskless House members scofflaws? Israel, Democrats and the problem of the Middle East Joe Crowley to register as lobbyist for recording artists MORE said in a statement Wednesday.

Obama, meanwhile, has spent the past week barnstorming for a legislative package that would offset the sequester with a mix of additional tax revenues and spending cuts. In addition to a speech Tuesday at the White House where the president appeared flanked by first responders who could see furloughs or layoffs if the sequester hit, Obama taped eight local television interviews and has invited urban radio hosts to the White House on Thursday.

White House press secretary Jay Carney has also pushed back against Republican attempts to pin the sequester on the president.

"If they had nothing to do with it and never liked it in the first place, why, as a Republican congressman has said, did they vote for it, overwhelmingly?" Carney said. "One hundred and seventy-one Republicans in the House voted for it, compared to I think 95 Democrats. Every Republican leader in the House voted for it, including Speaker Boehner. And on the day it passed, the Speaker of the House said, 'I got 98 percent of what I wanted and I’m pretty pleased.' "

But while Thursday's poll shows voters moving toward backing the White House narrative, Republicans have reason for optimism.

Seven in 10 say that it is essential for the president and Congress to pass major deficit legislation this year, higher than any of Obama's other second-term policy priorities. And if negotiators fail to strike a deal, four in 10 Americans say it would be better to allow spending cuts to go into effect.

Still, congressional Republicans face a difficult task. While President Obama's approval rating has dipped — now at 51 percent, four points lower than his post-election high — he still outpaces the 25 percent approval rating that GOP legislators earn. And more than three quarters — 76 percent — of those surveyed say they want an eventual deal to include a combination of spending cuts and tax increases. By contrast, only 19 percent agree with GOP leaders, who have said that deficit reduction should only include spending cuts.

That said, 73 percent say efforts to reduce the deficit should be mostly focused on spending cuts, while fewer than two in 10 say the same about tax increases, evidence that the GOP has some ammunition in the looming budget battle.