On the road in the important electoral states of Virginia and North Carolina the past two days, Mr. Obama has joked that his administration is breaking up its $447 billion jobs bill into separate chunks so it would be easier for befuddled Republicans in the Senate to understand.

But the increasingly caustic tone of the president’s attacks on Congress raises a question: How long can Mr. Obama continue to hammer Republicans without exhausting the patience of voters who elected him to be an alternative to Washington partisanship — and without risking the perception that he is part of the problem?

As his bus rolled from the Blue Ridge Mountains toward the Virginia Piedmont, the president delivered an unwavering message: the nation is facing a jobs crisis of historic proportions, and Republicans are blocking his attempts to deal with it.

“Maybe they just couldn’t understand the whole bill at once,” Mr. Obama said to a chortling crowd at the airport in Asheville, N.C. “We’re going to break it up into bite-size pieces.”

Republicans seem likely to reject several parts of Mr. Obama’s package, including new money for retaining teachers and repairing schools. And the president’s acerbic tone suggests that he knows as much.

Running against a hostile Congress is an appealing strategy for a president who, however much he has slipped in the polls, is still far more popular with voters than are his foes on Capitol Hill. Polls also show that most voters support large parts of the jobs bill, a point Mr. Obama drove home at every stop.

But the approach is not without risks for the president.

“What he’s obviously trying to do is model himself on Harry Truman and the ‘do-nothing Congress,’ ” said David Winston, a Republican strategist. “The problem with that is that the unemployment rate in November of 1948 was 3.8 percent.”

While Mr. Obama’s partisan jabs appeal to his Democratic base, they may turn off independent voters, who flocked to him in 2008 in part because of his carefully cultivated image as a leader who rises above the partisan fray. With the jobless rate closer to 10 percent than 4 percent, they may start to tune out the president.

The risks and advantages of the strategy were on display Tuesday at Guilford Technical Community College in Jamestown, N.C., where Mr. Obama spoke about the first piece of the jobs bill to face a vote: $35 billion in federal aid to states and cities for teachers, firefighters and police officers.

Mark Jewell, a 47-year-old teacher trainer, said that North Carolina’s schools desperately needed that money, and that he backed the president’s strategy of forcing Republicans to declare their position on every component of the jobs bill.

“We’re going back 50 years because of a Congress that wants to vote against anything Obama does,” Mr. Jewell said. “The Republicans are using it as a campaign tactic.”

But Thomas O’Connell, a 20-year-old student at the University of North Carolina, Greensboro, said that blaming Congress was not enough. “If he’s going to say the system is broken, he’s got to put forward something of substance himself,” he said.

Mr. O’Connell, who described himself as an Obama supporter but a disenchanted one, said he wished the president would propose public financing of political campaigns, rather than raising $1 billion for his own campaign.

There was a lot of sympathy for the president’s predicament among those who lined up to hear him at the community college. But there was also a sense that people’s patience was limited.

“We want to stand behind him and support him, but at some point we also want to see forward motion,” said Natalie Hopkins, an administrator in the Guilford County school district.

White House officials argue that Mr. Obama has laid out a detailed list of remedies in the American Jobs Act, including school aid and help for veterans in finding jobs, which the president will highlight with the first lady on Wednesday in Virginia.

In the end, they argued, Republicans will support some of the measures. But with at least five votes on parts of the bill likely to come up in the Senate, the exercise could keep the spotlight on those lawmakers until Thanksgiving. “This is the first act of a long drama,” a senior White House official said last week.

Mr. Obama seemed to recognize the yearning for an end to the rancor. He spoke of his efforts to find common cause with Republicans, telling the crowd in heavily Democratic Asheville, “Some of you have been mad at me for trying too hard to cooperate with them, haven’t you?” And he praised a free trade agreement with South Korea that passed the Senate with Republican support.

He also acknowledged that the Republicans had come up with an alternative to his jobs plan, though he quickly dismissed it. Speaking at a high school here, Mr. Obama claimed that the package would gut environmental regulations, roll back Wall Street reforms, leave 30 million Americans without health insurance and perhaps even cost jobs.

“I don’t know how you present a plan for jobs that results in less jobs,” Mr. Obama said to titters. “I mean, they didn’t call it the American No-Jobs Act.”

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