Senate Select Committee on Intelligence chairwoman Dianne Feinstein, D-Calif., said the committee is reviewing the conditions and interrogation techniques. The study will likely be completed in six to eight months, she said.
House Speaker Nancy Pelosi, D-Calif., hinted that there would be multiple congressional hearings on the issue.
"I myself do not believe that immunity should be granted to everyone in a blanket way," she told reporters, adding that she is in favor of congressional hearings on the subject and the creation of a "Truth Commission," also proposed by the White House.
Senate Judiciary Committee chairman Sen. Pat Leahy, D-Vt., pledged that if he cannot get the votes to create a bipartisan commission to investigate the Bush administration's torture policy, he will conduct his own partisan inquiry in the Senate Judiciary Committee.
Leahy said Bybee was not forthcoming during his confirmation process and that he would never have been confirmed had the Senate been aware of the content of the memos. Leahy added that Bybee should resign.
There is an ongoing internal investigation into Bybee and two other authors of the torture memos, John Yoo and Steven Bradbury, within the DOJ.
House Judiciary Committee Chairman John Conyers, D-Mich., announced Tuesday that the committee would hold hearings on the torture memos.
"Critical questions remain concerning how these memos came into existence and were approved, which our committee is uniquely situated to consider," Conyers said in a statement Tuesday.
"The president's comments ... on possible approaches to a fuller accounting of these matters are exactly right -- further comprehensive review of the Bush administration anti-terror policies will be most valuable and successful if done in a truly apolitical and bipartisan manner," he said.
Sen. Russ Feingold, D-Wis., wrote a letter today to Obama urging a thorough Justice Department investigation of the interrogation policy and saying Obama should not rule out prosecuting the CIA agents who carried out the interrogations.
It is unclear whether the Obama administration will get a bipartisan panel to investigate the memos and officials, a move the president has said would be essential in this type of an investigation. But Republicans have bashed the Democrats and White House for releasing the memos and discussing possible prosecution.
"We have been monitoring this. The leaders of the House and Senate and the intelligence committees at the time were briefed on these techniques. Now, to go back and start penalizing or punishing a previous administration is what a banana republic does," Sen. Kit Bond, R-Mo., said on "Top Line." "Our country has never done that, and I think this is a terrible step in the wrong direction."
Even Sen. John McCain, R-Ariz., who serves on the Senate Armed Services Committee and is a long-time critic of the CIA's interrogation policies, joined with his Republican counterparts in urging Obama not to prosecute the authors of the so-called torture memos.
In a letter to Obama, McCain, Sen. Joe Lieberman, Ind-Conn., and Sen. Lindsey Graham, R-S.C., said prosecutions would have a negative effect on how officials provide their advice.
"We have every interest in looking forward to solutions, not backward to recriminations," they wrote. "That is why we do not support the idea of a commission that would focus on the mistakes of the past."