A day after debt-reduction talks led by Vice President Biden appeared to have broken down, the White House announced that President Obama would directly intervene in the negotiations, beginning meetings with Biden and key lawmakers next week.

Obama and Biden will start by meeting separately with Senate Majority Leader Harry M. Reid (D-Nev.) and Senate Minority Leader Mitch McConnell (R-Ky.) on Monday.

White House press secretary Jay Carney told reporters Friday aboard Air Force One that “we believe that we can move forward as long as no one in the talks takes a ‘my way or the highway’ approach.”

The parties face an Aug. 2 deadline set by the Treasury Department to raise the country’s $14.3 trillion debt limit before the country risks defaulting on its debt obligations.

“I have a larger vision for America . . . where we work together, Democrats and Republicans, to live within our means, to cut our deficit and debt, but also to invest in what our economy needs to grow,” Obama said Friday before a crowd at Carnegie Mellon University, where he announced a new initiative to spark job growth.

But a quick compromise seemed unlikely Friday as House Speaker John A. Boehner (R-Ohio) reiterated his pledge to oppose any comprehensive deficit-reduction deal that would increase taxes. The issue of tax increases is one of the fundamental philosophical differences between the parties that congressional negotiators just this week began to address in the Biden talks.

“The president and his party may want a debt limit increase that includes tax hikes, but such a proposal cannot pass the House,” Boehner said in a statement.

He said any deal must include spending cuts that exceed the amount of the debt-limit hike and budget reforms that will restrict Washington’s ability to spend in the future — not tax increases.

Conservative Sen. Jim DeMint (R-S.C.) upped the ante Friday, warning in an interview with ABC News that Republicans who vote for raising the debt ceiling without passing a balanced-budget amendment and sweeping spending cuts are “gone.”

“It would be the most toxic vote,” DeMint told ABC News. “I can tell you, if you look at the polls, Democrats, Republicans, independents, they do not think we should increase the debt limit.”

Congressional Republicans, who abruptly pulled out of debt-reduction talks with the White House on Thursday, said the negotiations reached an impasse because Democrats were pressing what Boehner called “job-killing tax hikes.”

The White House pushed back Friday, saying the administration is seeking fresh revenues that target corporations and the wealthy, not small businesses and the middle class.

People familiar with the discussions said the White House has offered a menu of revenue raisers that would generate between $200 billion and $300 billion over the next decade. They include a variety of corporate provisions, such as an adjustment in the way inventory is taxed that would generate about $70 billion, higher tax rates on what is known as “carried interest” earned by hedge fund managers that would generate about $20 billion, and an end to a perk that saves the owners of corporate jets about $3 billion over 10 years.

The White House is also considering a fee on financial transactions that would generate about $30 billion over the next decade.

For individuals, administration officials are looking at a plan promoted by House Democrats that would cap deductions for households earning more than $500,000 a year at 10 percent of adjusted gross income. That provision alone could generate as much as $210 billion over the next decade, depending on the details.

“We’re focused on millionaires and billionaires,” said a senior administration official, speaking on the condition of anonymity to discuss the talks. “We want things that are unjustifiable loopholes, are indefensible on economic grounds and are indefensible on fairness grounds. Carried interest and corporate jets meet both of those tests.”

Administration officials declined to speculate when talks between the White House and both chambers of Congress will resume in earnest. But they emphasized that much progress has been made in the Biden talks over the past seven weeks, and that there is reason to be optimistic that the Aug. 2 deadline will be met.

“The sooner people can resume progress the better,” the official said. “I don’t see any reason to doubt that we’ll be able to do a long-term extension of the debt limit accompanied by very serious deficit reduction.”

Biden and six lawmakers from both parties had tentatively agreed to more than $1 trillion in savings and had begun to tackle the toughest issues: Democratic demands for higher taxes and spending cuts at the Pentagon, and Republican demands for sharp cuts to health and retirement programs.

“It is my hope that the President requested this meeting in order to finally explain what it is that he’s prepared to do to solve our nation’s fiscal crisis,” McConnell said in a statement.

Speaking at an event in Richmond on Friday, Sen. Mark R. Warner (D-Va.) accused the Republicans of “playing political games.”

Warner and Sen. Saxby Chambliss (R-Ga.) have been leading a bipartisan band of senators, dubbed the Gang of Six, to forge a comprehensive deficit-cutting plan — a mix of spending cuts and changes to the tax code and entitlement programs.

This problem can’t be solved “on one side of the balance sheet” by just cutting spending or raising taxes, Warner said. “The numbers don’t lie.” He said he hopes to resolve the last few differences and unveil the Gang of Six plan in a week or so.

Staff writer Anita Kumar and Zachary Goldfarb contributed to this report.