"The president has made clear: We cannot negotiate about whether or not the government of the United States would default," he told "Fox News Sunday," one of four talk shows on which he appeared.
"It was a mistake in 2011 to have that debate. It's hurt the economy. And I think Congress knows that it has to deal with this," he said.
But congressional Republicans have said they won't raise the debt ceiling, which is the government's statutory borrowing authority, without concessions. Those concessions include changes to entitlement programs such as Medicare, withholding funding for, if not repealing, Obama's signature healthcare-reform program and creating a mandatory timeline to lower tax rates through a tax-code overhaul.
"We're not going to raise the debt ceiling without real cuts in spending. It's as simple as that," Speaker John Boehner, R-Ohio, said Tuesday.
Senate Majority Leader Harry Reid, D-Nev., has said the Senate will not go along with the proposed House cuts.
The fight is expected to intensify when Congress returns from its August recess. After the Sept. 30 deadline for a spending bill to keep the government running, lawmakers face an early November deadline to raise the debt limit or risk U.S. default.
"What the president did this week is he sent a very clear message that Washington has to stop playing these brinkmanship games," Lew told NBC's "Meet the Press."
"It is imperative that Washington be part of the solution, not part of the problem," Lew, who previously served as Obama's chief of staff and budget director, told ABC's "This Week." "We can't afford self-inflicted wounds and we can't have these kinds of self-created crises month after month, year after year."
Sen. Mike Lee, R-Utah, who is leading the charge to de-fund the Patient Protection and Affordable Care Act, also known sometimes pejoratively as Obamacare, insisted the budget fight wouldn't lead to a shutdown.
"We all know that the government is going to get funded," he told "Fox News Sunday." "The only question is whether the government gets funded with Obamacare or without it."
Boehner's office challenged Obama last week to stop the shutdown talk by dropping a veto threat to the House budget bill, which continues current automatic, across-the-board spending cuts known as sequestration but shifts the cuts from the Pentagon to domestic programs.
"That's unacceptable. He won't sign that," Lew told "Meet the Press."
The current sequestration -- which was put in place during the last debt-ceiling crisis in 2011 as part of the Budget Control Act -- splits the cuts equally between defense and non-defense programs..
"Congress needs to get its work done and needs to fund the kinds of things the American middle class need, and we need to get the debt limit extended in a way that doesn't create a crisis," he said. "That is what every [member of] Congress needs to do, and Congress needs to do it when it gets back in September."