President Obama formally entered debt talks Monday with a pair of Oval Office meetings with Senate leaders, hoping that face-to-face talks could set the stage for detailed negotiations seeking more than $2 trillion in federal savings in exchange for continued Treasury borrowing to finance government operations.

With just five weeks to go before the federal government would begin defaulting on some of its debts, Obama hosted Senate Majority Leader Harry M. Reid (D-Nev.) and Senate Minority Leader Mitch McConnell (R-Ky.) in separate hour-long meetings.

The leaders remain divided over hundreds of billions of dollars in proposed spending reductions and also the key issue of whether to include increased tax revenue to fill up some of the savings.

While no immediate breakthroughs were reported, the talks were meant to signal Obama’s increased personal involvement in the issue after the dissolution last week of a group led by Vice President Biden and congressional leadership deputies. With anywhere from $500 billion to more than $1 trillion in spending cuts tentatively agreed upon, Obama and the top leaders will now work to iron out the most contentious issues left: tax increases, entitlement spending and Pentagon funding.

“Neither party should confront this crisis alone, and no one will be successful unless we confront it together,” Reid said in a speech on the Senate floor after his late-morning sit-down with the president. Reid wants a deal that would include new investments to jump-start job creation and close tax loopholes and subsidies that benefit the wealthy.

White House press secretary Jay Carney reiterated Obama’s call for a “balanced’’ approach focusing on increasing revenue into the Treasury through adjustments to the tax code. “It’s the only way to get it done if you want to do it right,’’ Carney said.

Those proposals, however, drew a sharp rebuke from McConnell before he joined Obama for a 5 p.m. meeting, insisting that tax increases and new spending could never pass Congress and calling on Obama to lay out a strategy for reaching a compromise that includes neither.

“Let’s move past the tax hikes and talk about what’s actually possible,” McConnell said. “This much is clear: If government spending were the answer to an economic slowdown, we’d be in a boom time right now.”

Congress must approve an increase in the current $14.3 trillion debt ceiling by Aug. 2, or else, according to Treasury Secretary Timothy F. Geithner, the nation will begin to default on its outstanding debts. Hoping to avert such a financial crisis, Biden’s group had been trying to craft more than $2 trillion in savings in exchange for a congressionally approved increase in the debt ceiling by a similar amount. A more than $2 trillion increase in the debt ceiling would allow the Treasury to borrow sufficient funds to finance government operations through 2012.

Assembling that level of savings, however, has become a game of legislative Whac-a-Mole: For every proposal that brings in new revenue or spending reductions, opponents pop up objecting to those ideas and making the political math toward rounding up the votes just as murky.

According to those involved in the talks, the White House has offered a roughly $1.3 trillion mix of cuts to domestic and national security agencies, but congressional Republicans had been pushing for $1.7 trillion. But Sen. Jon Kyl (R-Ariz.), who participated in the Biden group, suggested Monday that it had really only identified $500 billion in savings that all sides supported.

How the negotiators reach the $2 trillion line has set off an ideological battle across the Capitol.

Some Republicans have demanded that a stringent balanced budget amendment be attached to the debt-increase legislation. House Democrats vigorously oppose such a plan. Other GOP leaders have pushed aggressively to include steep savings from Medicare, and Sen. Joseph I. Lieberman (I-Conn.), along with fiscal hawk Sen. Tom Coburn (R-Okla.), plans to introduce legislation Tuesday that would shave about $500 billion from Medicare’s coffers over 10 years, in part by increasing the eligibility age from 65 to 67.

Congressional Democrats have vowed to block any plan if it includes reduced benefits for seniors.

Sen. Bernard Sanders, a Vermont independent with socialist leanings, delivered a 90-minute address Monday outlining his plan calling for 50 percent of all savings to come from tax increases. “The wealthiest Americans and the most profitable corporations in this country must pay their fair share,” Sanders wrote Monday in a letter to Obama.

Such a proposal has no chance of passing because Republicans and many Democrats believe steep tax increases are both politically unpopular and potentially harmful to the struggling domestic economy.

Staff writer Felicia Sonmez contributed to this report.