Amash introduced an amendment in the House to curtail the NSA's bulk collection of U.S. phone records, but the amendment narrowly failed.
Meanwhile, in an interview published Thursday with Rolling Stone, Sen. Ron Wyden, D-Ore., said, "I'm not sure those at the highest level have really come to see the implications of their not being straight with the Congress."
On Friday, Wyden and Sen. Mark Udall, D-Colo. -- both members of the Intelligence Committee -- released a joint statement calling on the administration to release more details of the violations.
"We have previously said that the violations of these laws and rules were more serious than had been acknowledged, and we believe Americans should know that this confirmation is just the tip of a larger iceberg," they said.
Mark Jaycox, a policy analyst with the Electronic Frontier Foundation, told CBSNews.com, "This is exactly what's not supposed to happen -- a rogue agency keeping documents secret from the people who are supposed to oversee them."
That said, he added, "Congress has not been given enough flak" for failing at its oversight duties.
"The congressional intelligence committees have not provided the oversight they were originally intended to," he said, suggesting that the House and Senate Judiciary Committees launch an investigation or that an independent committee like the 9/11 Commission investigate surveillance activities.
Senate Judiciary Committee Chairman Patrick Leahy, D-Vt., said in a statement in response to the Post's report that he plans to hold another hearing on the matter. He noted that he's re-introduced legislation to strengthen independent oversight of the NSA, in part by requiring an inspector general audit -- not just an internal audit -- of surveillance activities.
"Every one of us who reads the intelligence knows we face threats, we face horrendous threats, but there are better ways of blocking those threats and the idea that we're going to just spy on every single American and do things that we never believed in as a country is not going to make us safer in the long run," Leahy said in an interview with Vermont CBS affiliate WCAX. "I want to know if NSA has made a mistake, we ought to know that. If they are tapping into people's telephones where they have no right to, we ought to know that. And if they are spending huge amounts of money to collect data, but not doing anything to make us safer we ought to know that."
While Congress has a responsibility to conduct oversight of surveillance activities, it's also the job of the Foreign Intelligence Surveillance Court (FISC). Yet in a complementary article, the Washington Post makes clear the court's powers are limited.
"The FISC is forced to rely upon the accuracy of the information that is provided to the Court," U.S. District Judge Reggie B. Walton, chief of the FISC, said in a written statement to the Post. "The FISC does not have the capacity to investigate issues of noncompliance, and in that respect the FISC is in the same position as any other court when it comes to enforcing [government] compliance with its orders."