With many voters expressing growing alarm at the mounting national debt, Republicans say that standing against an unemployment extension that would add to the deficit could energize their voters and help them regain some of the reputation for fiscal responsibility they have lost in recent years. They also accused the White House of misleading the public about the Republican position on added jobless pay.

“The president knows that Republicans support extending unemployment insurance, and doing it in a fiscally responsible way by cutting spending elsewhere in the $3 trillion federal budget,” Representative John A. Boehner of Ohio, the House Republican leader, said in a statement Monday. “At a time of record debt and deficits made worse by Washington Democrats’ massive spending spree, that’s the right thing to do and the right way to do it.”

The additional money for those who have exhausted their standard 26 weeks of jobless pay has been tied up since the beginning of June but had become a growing point of contention since February when Senator Jim Bunning, Republican of Kentucky, initiated a one-man filibuster against a temporary extension of the safety-net spending.

Image Standing with three Americans who have struggled to find work, President Obama spoke in the Rose Garden about the need to extend unemployment benefits. Credit... Stephen Crowley/The New York Times

While Republicans eventually relented and allowed an additional month of unemployment compensation, the party began to coalesce around the position that further extensions should be paid for with offsetting cuts in other spending, leading to the current stalemate.

Most Democrats contend that deficit spending is acceptable  even, in economic terms, necessary  to help not only the jobless but also the economy as a whole. Their argument is that unemployed workers will spend all or nearly all of their benefits on goods and services that help support other jobs.

“At what point do we pivot and start being concerned about our children and our grandchildren?” Senator Mitch McConnell of Kentucky, the Republican leader, said Sunday on CNN. “There is no way in the world on a trillion-dollar budget this year we can’t find the money to pay for an extension of unemployment insurance, something we’re in favor of.”