Video: Courtesy of NBC

WASHINGTON — President Obama on Monday made an unscheduled visit to the Federal Emergency Management Agency in his latest bid to draw attention to the effects of the government shutdown and to challenge Speaker John A. Boehner’s claim that he does not have enough votes to pass a measure to finance the government.

Mr. Boehner, appearing Sunday on the ABC News program “This Week,” said the Republican-controlled House could not pass a budget measure without provisions to limit Mr. Obama’s health insurance law. Democrats and some Republicans say that a bipartisan majority exists but that Mr. Boehner refuses to defy Tea Party conservatives.

At FEMA, Mr. Obama called on Mr. Boehner to put to a vote the Senate-passed stopgap bill to fund the government.

“The House should hold that vote today,” Mr. Obama said. “If Republicans and Speaker Boehner are saying there are not enough votes, then they should prove it. Let the bill go to the floor and let’s see what happens. Just vote. Let every member of Congress vote their conscience and they can determine whether or not they want to shut the government down. My suspicion is, my very strong suspicion is, that there are enough votes there.”

Turning up the pressure, Mr. Obama added, “The reason that Speaker Boehner hasn’t called a vote on it is because he doesn’t apparently want to see the government shutdown end at the moment, unless he’s able to extract concessions that don’t have anything to do with the budget.”

Like other federal agencies, FEMA is operating with reduced staff — 86 percent was furloughed, according to its chief, W. Craig Fugate, though some were temporarily called back to deal with storm response efforts, and the rest are working for now without pay. The agency has been busy coordinating with state and local governments to prepare for and respond to violent storms in the Midwest and Tropical Storm Karen.

The president used the appearance at the agency to reiterate his now-daily calls for Congress both to vote immediately to finance the government in the new fiscal year, which began last Tuesday, and to extend the nation’s borrowing limit, which the Treasury Department has estimated will be breached Oct. 17. Last Friday, he did the same with an unscheduled appearance at a local sandwich shop.

While the shutdown has posed myriad problems nationwide for citizens and businesses, failure to raise the debt ceiling could ultimately provoke the nation’s first default and lead to a global financial crisis.

House Republicans have refused to allow a vote on a Senate-passed funding measure until Mr. Obama negotiates limits or cutbacks in his signature health care law, which he has refused to do. The House in recent days has passed a number of bills funding various popular functions of the government, including FEMA, as public complaints have mounted, but the White House and Democratic-controlled Senate have rejected that piecemeal approach, demanding that the entire government be funded.

Mr. Boehner and other Republicans have excoriated Mr. Obama for his refusal to negotiate over their demands. Also at FEMA, Mr. Obama countered that line of attack.

“I have said from the start of the year that I’m happy to talk to Republicans about anything related to the budget,” Mr. Obama said. “There is not a subject that I am not willing to engage in, work on, negotiate and come up with common-sense compromises on. What I’ve said is that I cannot do that under the threat that if R’s don’t get 100 percent of their way they’re going to either shut down the government or they are going to default on America’s debt so that America for the first time in history does not pay its bills.”

“We’re not going to establish that pattern,” he added.

Separately, on Monday the president’s chief White House economic adviser, Gene B. Sperling, publicly reiterated what the Treasury secretary, Jacob J. Lew, has already told Congressional Republicans: the White House will let Congress decide how great an increase to provide in the nation’s soon-to-be-breached $16.7 trillion debt limit.

Mr. Lew, in a letter over the weekend to Senate Republicans, said that under the Constitution, “Only Congress can authorize an increase in the nation’s borrowing authority, and therefore Congress must choose how long to extend the debt limit.”

The Treasury secretary expressed a preference for bigger increase, which he did not quantify, to cover a longer time. “A longer period of certainty would help protect our economy from future political brinkmanship,” Mr. Lew wrote.