WASHINGTON—In a striking policy shift that ensured the rancorous spy-policy debate would extend into the fall, President Barack Obama announced plans to overhaul key parts the National Security Agency's surveillance programs.

The proposal—an extraordinary step in the face of this year's revelations from fugitive NSA leaker Edward Snowden—drew sharp responses from Republican lawmakers who suggested the president was retreating under political pressure. The tone of the reaction from Republicans and skeptical Democrats suggested Mr. Obama has a new fight on his hands, reminiscent of the two-year battle that preceded reforms of the surveillance law in 2008, during the George W. Bush administration.

The most significant proposal would restructure the Foreign Intelligence Surveillance Court, the secret court that oversees surveillance programs in the U.S., to provide for an advocate for privacy concerns. Mr. Obama is also seeking unspecified reforms to the Patriot Act to increase oversight and place more constraints on the provision that permits government seizure of business records.

Mr. Obama's actions came as officials anticipate continued revelations from Mr. Snowden, the NSA leaker currently sheltering in Russia. The two main programs revealed by Mr. Snowden are the vast collection of Americans' phone records and a set of court-ordered partnerships with Silicon Valley companies to provide Internet-account information for foreign-intelligence investigations.

The president acknowledged that Mr. Snowden's actions had sparked a "much more rapid" policy debate, which was Mr. Snowden's professed goal. "It's not enough for me, as president, to have confidence in these programs; the American people need to have confidence in them as well," Mr. Obama said.