Washington is growing increasingly jittery about the prospect of automatic spending cuts decimating federal agency budgets in January. So when President Obama declared in Monday’s debate that the cuts “will not happen,” people took note.

“Believe me, when he said that, I jumped to my feet,” said Marion Blakey, president of the Aerospace Industries Association, which projects that the Pentagon cuts alone would destroy a million jobs. “I think Americans hope he meant what he said.”

Minutes after the debate ended, however, senior advisers were playing down Obama’s assertion. On Tuesday, White House spokesman Jay Carney argued that the president had said nothing new. And while Republicans and Democrats sparred over the political implications of Obama’s remarks, their practical effect on the budgetary time bomb set to blow after the election appeared to be nil.

“We were pleased by the president’s comments initially,” said Dorothy Coleman, vice president of tax policy for the National Association of Manufacturers. “But promises have to turn into actions.”

If anything, the incident served to demonstrate how starved people are for information about how policymakers plan to deal with the “fiscal cliff” at the end of the year, which threatens to slice $100 billion out of agency budgets and raise taxes by about $400 billion next year. Unless Congress acts to stop it, these spending cuts and tax hikes are likely to drag the economy back into recession, according to official projections.

Yet the cliff has been virtually ignored on the campaign trail. The exchange Monday night in Boca Raton, Fla., marked the first time in three televised debates that Obama and Republican challenger Mitt Romney have addressed the issue that might pose the biggest immediate threat to the economy.

Blakey said the oversight made her “extremely impatient.” But she didn’t learn much when Romney finally tackled the issue, accusing Obama of proposing to “cut our military budget by a trillion dollars, which is the combination of the budget cuts the president has [endorsed], as well as the sequestration cuts.”

Obama fired back, saying: “First of all, the sequester is not something that I proposed. It’s something that Congress has proposed. It will not happen.”

After the debate, White House senior adviser David Plouffe argued that Obama was merely stating something that everyone agrees with: “No one wants it to happen.”

“The sequester . . . was never meant to become policy. It was never meant to be implemented. It’s a designed trigger, a forcing mechanism to compel Congress to make the difficult decisions required to reach a balanced deficit-reduction package,” Carney said.

Carney noted that the “Republicans, especially in the House,” insisted on an all-cuts trigger during last year’s budget battle, while the White House advocated a trigger that included automatic tax increases. Republican opposition to higher taxes on “millionaires and billionaires,” Carney argued, remains the biggest obstacle to avoiding the fiscal cliff.

But Obama has thrown up a few obstacles of his own. Late last year, he threatened to veto any deal that tries “to get rid of these automatic spending cuts” without replacing them with a more rational deficit-reduction plan. He has also threatened to veto any deal that extends expiring tax cuts for the wealthy, a top GOP priority.

Last month, Treasury Secretary Timothy F. Geithner repeated Obama’s demand for higher taxes on the wealthy in separate meetings with House Speaker John A. Boehner (R-Ohio) and House Ways and Means Chairman Dave Camp (R-Mich.), according to aides in both parties. Obama’s allies in Congress say that they expect him to play hardball on the issue if he wins reelection Nov. 6, and some Republicans are already predicting that the GOP could cave on the issue.