Mr. Obama’s senior adviser, David Plouffe, and his chief of staff, William M. Daley, want him to maintain a pragmatic strategy of appealing to independent voters by advocating ideas that can pass Congress, even if they may not have much economic impact. These include free trade agreements and improved patent protections for inventors.

But others, including Gene Sperling, Mr. Obama’s chief economic adviser, say public anger over the debt ceiling debate has weakened Republicans and created an opening for bigger ideas like tax incentives for businesses that hire more workers, according to Congressional Democrats who share that view. Democrats are also pushing the White House to help homeowners facing foreclosure.

Even if the ideas cannot pass Congress, they say, the president would gain a campaign issue by pushing for them.

“The president’s team puts a premium on being above the partisan fray, which is usually the right strategy,” said Senator Charles E. Schumer of New York, the No. 3 Democrat in the Senate. “But on this issue, when he knows what the right thing to do is, and when a rather small group on one side is blocking any progress, you have to be willing to call that group out if you want to get anything done.”