"Every day this man is a fugitive, on the run, he puts the United States and national security more at risk," House Homeland Security Committee Chairman Michael McCaul, R-Texas, told "Fox News Sunday."
"The idea that he's in Russia -- I'm sure the Russians love this," he said "They're making a mockery of the United States foreign policy and I'm sure every day, they're extracting more and more information from this man."
He didn't say why he believed that was the case.
Russian President Vladimir Putin has said Russian intelligence agencies were not "working with" Snowden -- an assertion met with wide skepticism in the West.
Snowden, a former National Security Agency systems analyst who provided secret documents to British newspaper The Guardian, The Washington Post and other newspapers, said Friday he intended to remain in Russia for some time while seeking asylum elsewhere.
He is reported in the transit zone of Moscow's Sheremetyevo International Airport. He arrived in Moscow June 23.
The documents Snowden leaked revealed NSA domestic and international electronic surveillance, including logging Americans' phone calls and intercepting foreigners' Internet communications, with help from U.S. companies, through a program called PRISM.
When asked how far the White House should push the Kremlin to get Snowden back, McCaul said, "I think we should be putting any and all pressure we have -- economic, trade.
"I think the restart button with Putin was a mistake," he said, referring to a symbolic "reset" button the two countries pressed in March 2009 as part of a "fresh start" in U.S.-Russia relations.
"Clearly, it's not working in this case, because we have no diplomacy leverage over Russia," McCaul said. "They're thumbing their nose at the United States.
"And that's not diplomacy through strength. It's been the pattern of this administration to try to negotiate without strength. And I think now we're seeing the repercussions of that strategy."
Washington has asked Moscow to hand over Snowden to U.S. authorities, but Moscow says Snowden did not go through Russian immigration and customs enforcement and is therefore not legally in Russian territory.
But International Counter-Terror Training Association President Josef B. Linder in Moscow was quoted by the Izvestia newspaper when Snowden arrived as saying the fact Snowden flew to Moscow on an Aeroflot Russian airline meant under international law he was considered in Russian territory.
Putin has said Snowden is free to leave Russia and should do so.
The 30-year-old computer expert faces two charges under the 1917 U.S. Espionage Act -- unauthorized communication of national defense information and willful communication of classified intelligence to an unauthorized person -- along with charges of theft of government property.
U.S. President Barack Obama and Putin discussed the Snowden case by phone Friday, the White House said, without giving details.
McCaul said the Snowden incident was "a test of our foreign policy, our diplomacy.
"I would give the administration low marks for what they've done so far," he said.