Senate Minority Leader Mitch McConnell outlined potential changes to Medicare and Social Security in an interview Friday, providing fresh clarity on the concessions Republicans would like to see from Democrats on cutting the costs of the federal entitlement programs.

Mr. McConnell (R., Ky.) said bipartisan agreement on higher Medicare premiums for the wealthy, an increase in the Medicare eligibility age and slowing cost-of-living increases for Social Security could move both parties closer to a budget deal that averts the so-called fiscal cliff, the combination of spending cuts and tax increases that start in early January unless Washington acts.

In return for the support of Democrats, he said, Republicans would agree to include more tax revenue in a budget deal, though not from higher rates.

Falling Over the Fiscal Cliff See some scenarios for how different groups of people may be affected by the tax changes that will take place if the fiscal cliff isn't resolved by the Jan. 1., 2013, deadline. More photos and interactive graphics

"Those are the kinds of things that would get Republicans interested in new revenue," Mr. McConnell said.

Democrats played down Mr. McConnell's comments and framed the debate from their own point of view: If Republicans instead agreed to raise income-tax rates for high earners, a deal to avoid the fiscal cliff could be quickly reached.

House Minority Leader Nancy Pelosi (D., Calif.) said there was "nothing new" in Mr. McConnell's comments. A senior administration official said the White House would make no new offers until Republicans changed their opposition to raising top tax rates.

Democrats said they were still awaiting a formal GOP proposal. "Republicans are still choosing not to put forward an actual offer, and we can't respond to an interview," said Adam Jentleson, spokesman for Senate Majority Leader Harry Reid (D., Nev.).

Mr. McConnell's cost-saving suggestions, however, mark a clearer articulation of the compromises his side was demanding.

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Republicans have said they would agree to revenue increases if Democrats went along with proposals intended to put safety-net programs on sounder financial footing. Democrats have countered that Republicans have been vague about what they want.

Mr. McConnell on Friday resurrected suggestions that were on the table during deficit-cut talks between Mr. Obama and House Speaker John Boehner (R., Ohio) in the summer of 2011, negotiations that broke down in finger-pointing on both sides.

Talks to avoid the fiscal cliff, which government economists say could throw the U.S. economy into recession, were at "stalemate," House Speaker John Boehner (R., Ohio) said Friday. Still, he said, talks had not collapsed and added, "I'm willing to move forward in good faith."

Senate Finance Committee Chairman Max Baucus (D., Mont.) said it was premature to expect major compromises. "It's too early," he said. "It's 31 days away."

Tensions between the two sides increased Thursday when Republicans dismissed a White House proposal that included $1.6 trillion in new taxes and $50 billion in infrastructure spending, among other things.

Mr. McConnell reiterated his rejection of that plan in Friday's interview, saying he laughed when Treasury Secretary Timothy Geithner presented it. "He noticed that I laughed," Mr. McConnell said. "That pretty well summed up my view of what he was saying."

The White House defended its offer. "This is the approach that garnered the president a lot of support in the election," said White House spokesman Josh Earnest. He said the president's proposal shouldn't have come as a surprise to Republicans.

On taxes, Mr. McConnell repeated Friday the GOP's position that any new revenue should come from capping deductions, not raising rates. "It is revenue and it's from the group of people they want to get revenue from," he said, arguing for Democratic support.

Mr. McConnell said his ideas for saving money on entitlement programs were examples of the structural changes sought by Republicans. "Not another study," he said. "Not 'Let's do it 11 months from now.' Now."

Senate Minority Leader Mitch McConnell, shown earlier this week, said in an interview Friday he wanted changes to safety-net programs that focus on altering eligibility requirements. Getty Images

Some Democrats have said they were open to more so-called means testing in Medicare, which would charge higher premiums to higher-income recipients. They have been more resistant to increasing the eligibility age from 65, and many Democrats have ruled out any changes to Social Security during the current fiscal talks.

While the two sides traded barbs in Washington, Mr. Obama kicked off a campaign intended to pressure Republicans, with appearances around the U.S. He repeated calls for Congress to extend the Bush-era tax cuts for families making less than $250,000 a year to workers Friday at a factory in Hatfield, Pa., that makes K'NEX construction toys.

"I want the American people to urge Congress soon—in the next week, the next two weeks—to begin the work we have by doing what we all agree on," Mr. Obama said. "I want you to call; I want you to send an email, post on their Facebook wall." Next week, Mr. Obama plans to meet with governors and deliver a speech to the Business Roundtable, a Washington lobby group.

Some analysts have suggested that serious talks between the parties were unfolding behind the scenes, even as Democrats and Republicans squared off in public. Mr. McConnell said Friday, "I can tell you there is nothing better going on privately."

The Obama administration has proposed making it difficult for Congress to block increases in the federal debt ceiling, a running source of political friction. The U.S. lost its pristine credit rating in 2011 after Republicans and the White House sparred until the last minute over a deal to raise the limit.

Republicans have called that proposal a presidential power grab. Sen. Charles Grassley (R., Iowa) said, "I don't know if that's going to be the straw that broke the camel's back, but it could be."

—Carol E. Lee and Andrew Ackerman contributed to this article.

Write to Naftali Bendavid at naftali.bendavid@wsj.com and Janet Hook at janet.hook@wsj.com