Instead, Obama is looking for a deal that includes a broader “framework” for future cuts, with the details to be negoiated during talks on the fiscal 2012 budget. Any deal, he has said, will also include his proposal to create an automatic trigger to impose tax increases and program reductions if the parties fail to reach future budget-cutting benchmarks.

“The most likely outcome,” predicts Peter Orszag, Obama’s former budget director, “is a series of targets backed up by a trigger.”

Democrats also hope to exploit what they see as a rift between House Speaker John Boehner (R-Ohio), who repeatedly has said the limit needs to be raised, and his No. 2, Rep. Eric Cantor (R-Va.), who has taken a tougher line, in keeping with his alliance with tea-party conservatives.

“We have the leverage here,” said Rep. Barney Frank (D-Mass.). “There are obviously divisions in the GOP among the leaders and the people pursuing their dumb notions of linking [cuts] to the debt vote.