The same day Russia allowed intelligence-leaker Edward Snowden to seek sanctuary inside the country, a group of Democratic senators began a push for changes to a secret U.S. court that approves government surveillance requests for the very programs Snowden exposed.

Democratic Sens. Ron Wyden, of Oregon, Richard Blumenthal, of Connecticut, and Tom Udall, of New Mexico, on Thursday unveiled two bills that would make changes to the Foreign Intelligence Surveillance Act (FISA) court system and change the way its judges are appointed.

The first bill would create a special advocate with the power to argue in the FISA court on behalf of the right to privacy and other civil liberties the Constitution is supposed to protect for the American people. Right now FISA court judges only weigh the arguments from one side – the government – which always favor surveillance programs.

The second measure would change the way judges are appointed to the FISA courts to ensure that the court is geographically and ideologically diverse and “better reflects the full diversity of perspectives on questions of national security, privacy and liberty,” Blumenthal said.

“This court is an anomaly” in the United States in that it exercises “vast invisible powers,” Blumenthal told reporters. “The Constitution more than ever needs a zealous advocate…decisions are made better when both sides in an argument are represented.”

In the court’s 33-year history, it has only turned down only 11 of the government’s 34,000 surveillance requests, Udall noted.

Under the measure, the presiding judge of the FISA Court of Review would appoint a special advocate for a five-year term from candidates nominated by a board created to promote privacy and civil liberty protections. In addition to advocating to minimize the scope of surveillance programs, the advocate would be able to appeal any court decision.

The court now hears only from the government and there is no process for appeal.

“How do you appeal something that’s secret?” Wyden asked, who said the changes would “unstack the deck” that now favors the government.

“This is a critical piece of the reform puzzle that will ensure that Americans maintain their security and their privacy,” he added.

Intelligence agency officials on Wednesday told a Senate committee that they’re willing to consider having a privacy advocated in the FISA process. Senate Intelligence Committee Chairwoman Dianne Feinstein, D-Calif., defends the vast surveillance programs as necessary to thwart potential terrorist attacks, but said she’s also open to creating a privacy advocate to argue against the government.

Several lawmakers, including Wyden, are heading to the White House later Thursday to meet with President Obama to discuss potential changes to the FISA process.