The California Democrat said she doesn't see Snowden as the hero some supporters view him, The Hill reported.
"I don't look at this as being a whistle-blower. I think it's an act of treason," said Feinstein, chairwoman of the Senate Intelligence Committee.
Citing investigations, White House press secretary Jay Carney didn't mention Snowden by name during his daily media briefing.
"There has obviously been some news over the weekend. I will say at the outset that there is, obviously, an investigation under way into this matter," Carney said. "And for that reason, I am not going to be able to discuss specifically this individual or this investigation, nor would I characterize the president's views on an individual or an ongoing investigation."
Earlier Monday, House Majority Leader Eric Cantor told CBS News lawmakers would conduct a "very serious" investigation into the leaked material.
Cantor said Obama administration officials would brief lawmakers Monday on how information about the cellphone and Internet surveillance program known as "Prism" was leaked to a British newspaper by Snowden, 29, a Booz Allen Hamilton contractor working with the NSA. Cantor said a more expansive briefing on how the NSA program works is scheduled for Tuesday.
Though pressed to comment -- Snowden said the reason for the leak was because he was disappointed with President Obama and thought the president didn't live up to his pledge to be transparent -- Carney said: "I'm not going to comment on the specific case or an individual. I think that the president's record on transparency is broad and significant."
On the "We the People" section of WhiteHouse.gov, more than 27,000 signatures were collected by mid-afternoon Monday on a petition asking Obama to pardon Snowden, who revealed the NSA's surveillance program known as Prism to The Guardian, a British publication. A petition needs 100,000 signatures to be considered by the White House.
"Edward Snowden is a national hero and should be immediately issued a full, free, and absolute pardon for any crimes he has committed or may have committed related to blowing the whistle on secret NSA surveillance programs," the petition read.
Stephen M. Kohn, executive director of the National Whistleblower Center, said in a release Snowden should not be prosecuted, as some in Washington indicated could happen, pending investigations undertaken by Congress and the administration.
"Until Congress enacts a law setting forth reasonable procedures by which civil servants can disclose national security violations to the American people, the government should not prosecute these whistle-blowers. Congress and the president must do their jobs, and stop destroying the lives of civil servants who try to report misconduct," Kohn said.
Carney said he didn't think the matter would affect any relationship-building Obama may try.
Obama "believes that this is a conversation especially worth having and a debate especially worth having here in the United States, but obviously beyond, as well," Carney said.
When considering Section 702 of the Foreign Intelligence Surveillance Act, Carney said it's "entirely appropriate for a program to exist to look at, you know, foreign data and ... potential foreign terrorists."
Carney recalled Obama's remarks Friday, saying the president "spoke fairly expansively about both his concerns and his belief that we need to strike the appropriate balance between our national security interests and our interests in privacy. ... And he believes that, with the oversight that exists and the implementation of the programs as they are implemented, that the balance is appropriately struck, has been appropriately struck, but it is an absolutely appropriate topic for debate, both now and going into the future, because the kind of technological advances we've seen when it comes to communications will only continue."
On CBS, Cantor said, "The investigations will be very serious. Obviously, we'll be dealing with a balance between national security and safeguarding our civil liberties."
Cantor said the House would begin the oversight process "to understand if there are laws that have been broken. Certainly, the reports seem to indicate that, that if anyone were to violate the law by releasing classified information outside the legal avenues, certainly that individual should be prosecuted at the full extent of the law."
Cantor said programs such as NSA's Prism are needed to help foil terrorist threats against the United States.
"I think that right now we know that there are active threats against the United States. We have terrorist threats [that] continue. There are possible security incidences that continue," the majority leader said. "And that is just the world we live in. We also know we have to balance the fact that we need to safeguard civil liberties."
Cantor said he and others were "perplexed" about the way Snowden released the information through the media -- specifically the British publication The Guardian -- instead of through methods established for whistle-blowers under U.S. law. The Guardian reported during the weekend Snowden fled to a Hong Kong hotel after revealing details about the NSA program.
The U.S. Justice Department says it has begun a criminal investigation of Snowden, saying in a statement it was in "the initial stages."
The Office of the Director of National Intelligence said the intelligence community was "reviewing the damage" from the leaks.