The top Senate Republican sought Thursday to clarify his party’s stance on Medicare heading into high-stakes talks with the White House, telling President Obama he wants “significant” changes to the program in exchange for lifting the legal limit on government borrowing.

After the entire Senate Republican caucus met with Obama at the White House, Minority Leader Mitch McConnell (Ky.) said he would not insist on a controversial House GOP plan that would partly privatize the popular health program for the elderly. But with Medicare and Medicaid projected to be the major drivers of future borrowing, he said tighter eligibility requirements and reduced benefits must be part of any deal.

“The things I’m talking about have already been studied to death. We don’t need any more hearings,” McConnell said at a Capitol Hill news conference. “We know what the options are. The only question remaining is what will we pick up and agree to on a bipartisan basis.”

McConnell repeatedly stressed that he was speaking “only for myself.” But he has emerged as a key legislative dealmakersince Republicans took over the House in January, maintaining close ties to House Speaker John A. Boehner (R-Ohio) while establishing a direct channel to the White House through Vice President Biden, his former Senate colleague.

“Speaker Boehner and Leader McConnell are working closely together” and “everything Senator McConnell discussed today” is consistent with the House GOP budget, Boehner spokesman Michael Steel said.

McConnell’s assistance is also critical to Senate Democrats. Unless they can win Republican votes, they would have to approve one of the largest debt-limit increases in U.S. history entirely on their own — an unappetizing prospect for up to a dozen potentially vulnerable incumbents who are up for reelection next year.

In recent days, McConnell has made clear that he is willing to cooperate with the White House in drafting a “grand bargain” to help push a debt-limit increase through a reluctant Congress, as long as it includes sharp cuts in both agency and entitlement spending. McConnell said he views the debt-limit debate as a critical opportunity for the parties to work together to accomplish something that would otherwise be impossible politically.

“If there is a grand bargain of some kind with the president of the United States, none of it will be usable for either side in next year’s election — none of it,” McConnell said. “We can do something important for the country together, and this is the opportunity.”

In addition to cuts to Medicare and Medicaid, McConnell called for an agreement to reduce spending at federal agencies over the next two years, a move that would defuse a battle over agency appropriations and lessen the risk of a government shutdown before the 2012 election.

McConnell said he also wants to see limits on spending set for 2014 and beyond, although he acknowledged that caps are often breached and are therefore a less reliable tool for debt reduction.

A major rewrite of the tax code, while popular with both parties, “will not be accomplished” as part of the debt-limit debate, nor will Republicans agree to tax increases of any kind, McConnell said. But he said he would not press Obama to reduce spending on Social Security, a major objective for three GOP senators working to draft a separate debt-reduction deal as part of the Gang of Six.

Obama’s meeting with Republicans came a day after a similar tete-a-tete with Senate Democrats and hours before a bipartisan group of lawmakers met for a third time with Biden in hopes of cutting a debt-limit deal. The national debt is set to hit the current $14.3 trillion ceiling within the next few days. Treasury Secretary Timothy F. Geithner has said he can keep paying the bills without a higher limit through Aug. 2, when the United States would run the risk of default.

Lawmakers in both parties say they are willing to raise the limit, but only in exchange for a plan to reduce future borrowing. The budget blueprint Obama sent to Capitol Hill in February would have required $9.5 trillion in fresh debt through 2021, and even the more austere House GOP budget would require nearly $2 trillion in new borrowing through the end of next year.

McConnell declined to say whether Republicans are willing to vote for a $2 trillion increase in the debt ceiling in one shot or prefer to vote multiple times for smaller increases. “How much and how long” is part of the negotiations, he said.

McConnell did not echo Boehner’s demand that the value of spending cuts exceed the value of the debt limit increase. But McConnell endorsed the Biden talks as the forum for achieving bipartisan compromise and said that “we will be recommending it to our members” if such an agreement is reached.

McConnell and other Republicans described their 75-minute meeting with Obama as “productive,” with several urging the president to tell Democrats to stop bashing the House GOP Medicare plan, which polls show to be deeply unpopular.

Sen. Bob Corker (Tenn.) said he told Obama: “Whenever we demagogue, all we do is push each other away from solving a problem we all know needs to be solved.”

Sen. Marco Rubio (Fla.) said he told Obama about his 80-year-old mother, a longtime Medicare recipient. “I want us to save Medicare with no changes for current beneficiaries and to ensure it exists for my generation and for my children’s generation. But we have to start dealing with it now,” Rubio said.

White House press secretary Jay Carney described the meeting as “constructive.”

“Anytime you can get Democrats and Republicans together, in this case a Democratic president and Republican senators together, to talk about this issue,” he told reporters, “that’s a positive thing.”