The House of Representatives has so far rejected the immunity, but the Senate Intelligence Committee approved a bill last month that would allow it. The Senate Judiciary Committee is set to consider that bill at a meeting Thursday.
Key Democrats and Republicans on the Senate Judiciary Committee have expressed concern with any sort of immunity, but not dismissed it.
Pennsylvania Sen. Arlen Specter, the ranking Republican on the committee, has suggested that Congress consider indemnifying telecom companies against damages.
"At least that would allow the lawsuit to go forward," EFF attorney Kevin Bankston said today.
President Bush has said he will veto any bill that does not include the immunity for telecom companies.
When the New York Times reported in late 2005 on the warrantless domestic wiretapping program run by the NSA, Klein, who had recently retired from AT&T, said he became "frustrated."
Klein linked up with the EFF in 2006 and is cooperating in their lawsuit.
"Its not the way a warrant should work that you take everything and sift away what you don't want," Klein said.
"Wiretapping in the past has always been a retail operation as opposed to a wholesale one. The government has had to determine who they want to target before they can target them," Bankston said.
In May 2006, Bush defended the NSA's warrantless programs by saying the government was not mining for data and only targeting foreign terrorists and al Qaeda operatives.
"First, our international activities strictly target al Qaeda and their known affiliates. Al Qaeda is our enemy, and we want to know their plans. Second, the government does not listen to domestic phone calls without court approval," he said.
"Third, the intelligence activities I authorized are lawful and have been briefed to appropriate members of Congress, both Republican and Democrat. Fourth, the privacy of ordinary Americans is fiercely protected in all our activities. We're not mining or trolling through the personal lives of millions of innocent Americans. Our efforts are focused on links to al Qaeda and their known affiliates. So far we've been very successful in preventing another attack on our soil," he said.
But Brian Reid, a former Stanford electrical engineering professor who appeared with Klein, said the NSA would logically collect phone and Internet data simultaneously because of the way fiber optic cables are intertwined.
He said the way the system described by Klein suggests a "wholesale, dragnet surveillance."
Bankston argued that simply by diverting the data, even if it did not look at specific messages, the government violates Fourth Amendment protections against search and seizure without probable cause.
Of the major telecom companies, only Qwest is known to have rejected government requests for access to data.
Former Qwest CEO Joseph Nacchio, appealing an insider trading conviction last month, said the government was seeking access to data even before Sept. 11.
Wednesday's press conference was conducted on Capitol Hill in the Senate Banking Committee's hearing room. The chairman of the Banking Committee, Connecticut Democrat Sen. Chris Dodd, who is also running for president, has said he will use procedural measures to block any legislation that offers retroactive immunity to telecom companies.