Congressional leaders on Sunday said a key part of President Obama's effort to overhaul U.S. surveillance will not work while another said the president didn't go far enough to protect Americans' privacy.

The leaders of both the House and Senate intelligence committees pushed back against the president’s assertion that the government should cede control of how Americans' phone records are stored.

Obama, under pressure to calm the controversy over government spying, said Friday he wants bulk phone data stored outside the government to reduce the risk that the records will be abused.

However, Obama did not say who should have control of Americans' data and instead has directed the attorney general and director of national intelligence to find a solution within 60 days.

Rep. Mike Rogers, R-Mich., chairman of the House Intelligence Committee, said that decision had intensified a sense of uncertainty about the country's ability to root out terrorist threats.

"We really did need a decision on Friday, and what we got was lots of uncertainty," Rogers told CBS’ “Face the Nation.” "And just in my conversations over the weekend with intelligence officials, this new level of uncertainty is already having a bit of an impact on our ability to protect Americans by finding terrorists who are trying to reach into the United States."

The president also said he will require a special judge's advance approval before intelligence agencies can examine someone's data and will force analysts to keep their searches closer to suspected terrorists or organizations.

"I think that's a very difficult thing," Sen. Dianne Feinstein, D-Calif., chairman of the Senate Intelligence Committee, told NBC’s “Meet the Press.” "The whole purpose of this program is to provide instantaneous information to be able to disrupt any plot that may be taking place."

Under the surveillance program, the NSA gathers phone numbers called and the length of conversations, but not the content of the calls.

Obama said the NSA sometimes needs to tap those records to find people linked to suspected terrorists. But he said the bulk data eventually should be stored somewhere out of the government's hands. That could mean finding a way for phone companies to store the records, though some companies have balked at the idea, or it could mean creating a third-party entity to hold the records.

Feinstein said many Americans don't understand that threats persist a dozen years after the Sept. 11 terrorist attacks.

"New bombs are being devised,” she said. “New terrorists are emerging, new groups, actually, a new level of viciousness. And I think we need to be prepared."

However, the lawmakers did praise the president for his defense of the National Security Agency's surveillance programs.

"First, I thought it was very important that the president laid out no abuses, this was not an illegal program, it wasn't a rogue agency," Rogers said.

The surveillance programs have been under fire since former National Security Agency analyst Edward Snowden absconded with an estimated 1.7 million documents related to surveillance and other NSA operations, giving the documents to journalists around the world. Revelations in the documents sparked a furor over whether Americans have been giving up privacy protections in exchange for intelligence-gathering on terrorism.

Sen. Patrick Leahy, D-Vt., chairman of the Senate Judiciary Committee, has proposed to end the NSA's bulk data collection program, putting him at odds with his fellow Democrat Feinstein on the issue.

Leahy suggested to “Fox News Sunday” that the chamber will continue to look at legislation to make its own changes and the final product will be impacted by Attorney General Eric Holder’s testimony Jan. 29  before the Judiciary Committee.

“There’s a concern that we have gone too much into Americans’ privacy,” the Democratic lawmaker told “Fox News Sunday.” “There’s still going to be legislation on this.”

But Leahy suggested he might not fight the president on allowing the NSA's surveillance programs to continue.

“I think we have a way that we could do this," he said. "The question is, what is Congress going to do on this? ... I just think that there should be an oversight. ... I believe in going after the bad guys. ... But you still have to have some checks and balances, or you have a government that can run amok."

The Associated Press contributed to this report.