President Obama told key members of Congress on Thursday that he was "open to suggestions" for reforming the National Security Agency surveillance programs that have embroiled his administration in controversy.

Ron Wyden, the Oregon Democrat who is among the Senate's leading critics of the NSA's bulk phone records collection, said he left a meeting at the White House confident that "constructive" changes to the programs would soon take shape.

"The president said, and I accept what he said, that he was open to suggestions," Wyden told the Guardian after the White House meeting. "And I smiled and said, 'You all are going to get a number of them from me."

Wyden said he "just plain out said" to Obama that the NSA's bulk ongoing collection of hundreds of millions of Americans' phone records "must end".

NSA officials have described the phone-records collection as a vital intelligence tool, though deputy director John C Inglis conceded in a Senate hearing on Wednesday that at most one terrorist attack was prevented because of the bulk collection during the program's seven-year history.

"My focus was on the urgency of reform," Wyden said. "The false choices that you could either have security or liberty, I don't think, is a case that the American people would accept."

Obama called the meeting with top legislators to address the upsurge in congressional discomfort with the surveillance, particularly the bulk phone records program. Last week, the House of Representatives came a few votes short of voting to end it, despite fierce White House and NSA lobbying.

But several lawmakers at the meeting signaled that they would fight to protect the bulk phone records and other NSA surveillance programs, despite the current uproar.

After the meeting, the Republican and Democratic chairs of the House and Senate intelligence committees said the meeting was "productive". They said in a joint statement: "We will continue to work through the August recess on proposals to improve transparency and strengthen privacy protections to further build the confidence of the American public in our nation's counterterrorism programs."

The meeting came a day after the Guardian revealed details of another National Security Agency surveillance program which, according to documents, allows analysts to to search through huge databases of emails, online chats and browsing histories without prior authorisation.

The training documents provided by the whistleblower Edward Snowden suggested that the program, called XKeyscore, is the NSA's "widest-reaching" system for developing intelligence from the internet.

Hours before the Thursday meeting, Snowden left the limbo of Moscow airport after being granted permission to stay in Russia for a year.

Almost two months on from the initial NSA disclosures in the Guardian and the Washington Post, based on documents leaked by Snowden, the White House appears to have entered a new phase as it seeks to head off criticism. The president has directed intelligence officials to find ways to make the NSA programs "as transparent as possible", while senior administration officials have testified before Congress that they are open to reforming the Foreign Intelligence Surveillance Court (Fisa), which issues orders for surveillance.

On Wednesday, the administration voluntarily declassified previously top-secret documents in an attempt to mollify critics.

Officials pointed to the documents to claim the surveillance programs are restrained, as rules imposed by the Fisa court require analysts to only search through phone data if they have "reasonable articulable suspicion" that the phone number is associated with terrorism or espionage. But on Wednesday, several senators on the judiciary committee objected during a contentious hearing that Americans' privacy is already compromised once the NSA collects their phone records in bulk without suspicion of wrongdoing.