Feinstein also called for legislation that would prevent government contractors such as 29-year-old Edward Snowden, who has admitted leaking the classified documents about the covert programs to the media, from having access to such sensitive information.
While saying he couldn't say much about a person under investigation, White House spokesman Jay Carney did tell reporters Thursday that "the leaks themselves were very serious."
"They go right to the heart of our efforts to combat terrorism, to combat efforts by extremists who desire to attack the United States and the American people," Carney said.
The White House spokesman also reiterated Obama's support for programs he says have "helped thwart dozens of attacks."
"He believes in the tradeoffs that we have to make to pursue our security and protect our privacy," Carney said of the president. "We have found ... we have the right balance, but (Obama) understands others may have a different opinion.
Like members of his national security team, the president has defended the Bush administration programs as necessary and claimed that there are more checks and balances on them than there were when he came to office in 2009.
Under his administration, legislative and judicial oversight of the Patriot Act has been strengthened, Obama told reporters last week.
Boehner, however, said Thursday that he was surprised the White House "hasn't stood up and made clear on an ongoing basis over this last week just how important these programs are."