When President Obama last spoke to the nation, he said he was ready to make Republicans an offer they couldn’t refuse. In exchange for trillions of dollars in cuts, including to Medicare and Social Security, Republicans would have to agree to a fraction of that in increased tax revenue.

“Even a deal that is not as balanced as I think it should be is better than no deal at all,” Obama said on Friday.

Obama’s political advisers have long believed that securing such an agreement would provide an enormous boost to his 2012 campaign, according to people familiar with White House thinking. In particular, they want to preserve and improve the president’s standing among political independents, who abandoned Democrats in the 2010 midterm elections and who say reining in the nation’s debt is a high priority.

In many ways, it has been a remarkable transformation for a Democratic president who had made the centerpiece of his first year in office a massive spending bill to boost the economy and the expansion of health insurance.

The risk for Obama now is that his pursuit of a far-reaching package could deeply disappoint his Democratic allies who believe he may be giving away too much. By calculating that an ambitious plan to reduce the nation’s debt by $4 trillion over 10 years is so important, he’s willing to endanger one of the best weapons in his party’s arsenal — the argument that Democrats will protect Medicare and Social Security at all costs.

“It’s a political and policy gamble — the idea that people would welcome a large deal when it required middle-class Americans to sacrifice, even down the road, at a time when they have fewer resources,” said Neera Tanden, chief operating officer at the Center for American Progress and former domestic policy adviser in the Obama administration.

Administration officials said the shift fits with Obama’s vision of what his presidency should look like.

“The president ran for the office to bring both parties together to solve big problems. That’s what he is trying to do here, even if it comes with political pain,” Dan Pfeiffer, White House communications director, said in an e-mail.

Obama has said a big deal is best for the nation’s economy and would help preserve entitlements over the long run. He has also started making the argument to liberals that they won’t be able to get their priorities accomplished until there is a plan to tame the debt.

What’s more, a failure to secure such a deal could prompt downgrades by credit-rating firms, which could lead to higher interest rates and would deal a fresh blow to the struggling economy.

“He wants to show himself to be the reasonable centrist, but if the economy’s bad, it’s not going to make any difference,” said Tom Davis, a former senior Republican lawmaker from Northern Virginia.

Several features of the offer Obama made to Republicans struck Democrats as conceding a great amount of ground, including raising the Medicare eligibility age from 65 to 67 over the long term and trimming Social Security benefits by making smaller yearly cost-of-living increases.

Obama also asked for $1.2 trillion in new tax revenue — equal to the Bush-era tax cuts for the wealthy, which are set to expire at the end of next year, and closing loopholes. The bipartisan Gang of Six group of senators called for $1.6 trillion in new tax revenues.

But when Republicans balked, Obama told House Speaker John A. Boehner (R-Ohio) that he could accept closer to $800 billion in revenue if entitlement cuts were also pared back.

Jared Bernstein, a former White House economics official and now a senior fellow at the Center for Budget and Policy Priorities, said that if the $800 billion in new revenues is “simply from the expiration of the high-end cuts, then it’s nothing more than expected anyway, and that victory would be a Pyrrhic victory.”

Unlike congressional Republicans and Democrats, who have to worry about what primary voters think of a deal when they’re up for reelection next year, Obama is focused on the views of independent voters in the general election. In polls, independents still show support for Obama, but his numbers have come down significantly since he took office.

“He sees the achievement of a big deal as a way to reach out effectively to the kinds of independent and moderate voters who supported him so strongly in 2008 and who have become somewhat disaffected in the interim,” said William Galston, a senior fellow at the Brookings Institution.

A deal would also help Obama inoculate himself against criticism from Republicans that he has failed to tame the debt.

“It benefits him politically a great deal if there’s a big deal,” said Jim Kessler, vice president for policy and a co-founder of Third Way, a left-leaning group that has been supporting an ambitious plan. “The deficit favors Republicans. But if it seems like it’s been solved, then you take away one of the biggest Republican issues and at worst you neutralize it and at best you win it.”