But Democrats said they would push for a "balanced" approach of more tax revenue from the rich as well as spending reductions as Congress headed toward another fiscal battle in March over raising the federal debt ceiling.
"The tax issue is finished, over, completed," McConnell, a Kentucky Republican, said on ABC's "This Week With George Stephanopoulos."
"That's behind us. Now the question is what are we going to do about the biggest problem confronting our country and our future, and that's our spending addiction."
McConnell used the Sunday morning news shows to lay out his position in the upcoming fight over raising the U.S. debt ceiling and funding the government that is expected to come to a head in March, just three months after the struggle to avert the January 1 fiscal cliff of severe tax hikes and spending cuts that economists said could have brought a recession.
Republicans want big spending cuts in programs including Medicare healthcare for the elderly and the Social Security pension program as a condition for raising the U.S. borrowing limit.
President Barack Obama has said he will not negotiate over the debt ceiling, arguing that Congress must pay the bills for spending it has already approved.
McConnell said the White House should start working with Congress immediately to find savings, before the March deadline to raise the borrowing limit brings another fiscal crisis.
"We could do things very quickly, these are not new issues," he said on ABC.
Asked whether Republicans would threaten a U.S. default in their press for spending cuts, McConnell said, "It's not even necessary to get to that point. Why aren't we trying to settle the problem? Why aren't we trying to do something about reducing spending?"
On CBS's "Face the Nation," he said, "We now have a debt of $16.4 trillion. That's as big as our economy. That alone makes us look a lot like Greece."
'LINE IN THE SAND'
Democrats said they will continue to push for more revenue as well as spending cuts to curb deficits, issues they said should be dealt with separately from the debt ceiling.
"Well, if Mitch McConnell is going to draw the line in the sand, it's going to be a recipe for more gridlock," Representative Chris Van Hollen, a Maryland Democrat, said on "Fox News Sunday."
"As we go forward, we need to adopt the same framework as the bipartisan Simpson-Bowles commission, meaning, a combination of cuts and revenue," Van Hollen said, referring to the commission that presented a sweeping plan to cut deficits.
"We're talking about looking at the tax code, putting everything on the table from the standpoint of closing loopholes, and we know that we can do that. Special subsidies for big oil, for example, $38 billion right there," House Democratic leader Nancy Pelosi said on CBS's "Face the Nation."
Pelosi, of California, said any savings found in Medicare and Social Security should be plowed back into the programs.
In what could be a replay of last year's standoff over the debt ceiling, House Republicans will put forward a plan "that says: OK, Mr. President, if you want to increase the borrowing authority of this country, here is a menu of options where you can reduce spending of equal or greater amount," said Ohio Republican Representative Jim Jordan.
"Mitch McConnell is exactly right," Jordan said on Fox News. "They just got revenue. We've got to cut spending. We've got $16 trillion debt. The credit card is maxed out."
Senator Dick Durbin, the No. 2 Democrat in the chamber, said Obama will discuss curbing the debt in his State of the Union address this month, "but it has to be done in a balanced way."
On CNN's "State of the Union," Durbin, of Illinois, said more money should be wrung from taxes, citing various deductions, special treatments and loopholes. "We can do that and use the money to reduce the deficit."
In his several television appearances, McConnell also defended the deal he helped to broker with Vice President Joe Biden to avoid the fiscal cliff.
Most of his fellow Republicans in the House opposed the deal for being focused almost entirely on raising revenue through a tax increase on families making more than $450,000 a year, while postponing significant spending cuts.
"What we did was prevent tax increases on 99 percent of the American public. Nobody in the Senate, not the 90 percent of Senate Republicans who voted for this, voted to raise anybody's taxes," McConnell said on NBC's "Meet the Press."
The deal extended lower tax rates for most taxpayers set during the George W. Bush administration that were set to expire on January 1, but let rates rise on the top incomes.
(Additional reporting by David Brunnstrom; Editing by Sandra Maler)