President Obama does not plan to meet one-on-one with Russian President Vladimir Putin as part of a trip to Russia next month if the asylum situation with NSA leaker Edward Snowden does not change.

Mr. Snowden, a former contractor for the National Security Agency, was granted temporary asylum by Russia last week. He had been holed up in the transit area of Moscow’s Sheremetyevo Airport after fleeing the U.S. and heading to Hong Kong in the wake of his disclosing vast amounts of information about U.S. surveillance programs.

After the G-20 summit in St. Petersburg next month, Mr. Obama had planned a bilateral meeting with Mr. Putin, but NBC reported Sunday on “Meet the Press” that the administration will announce this week the meeting will not take place if the asylum situation is not resolved. The Obama administration protested vigorously when Moscow granted the fugitive asylum last week, the latest in a string of irritations in the bilateral relationship.

Mr. Snowden has been charged with espionage after leaking information regarding government surveillance programs to news outlets in recent months.

Sen. Charles E. Schumer also said the meeting should be canceled, saying Mr. Putin was acting “like a schoolyard bully.”

“That [meeting] would give Mr. Putin the kind of respect he doesn’t deserve at this time,” Mr. Schumer, New York Democrat, said on CBS’s “Face the Nation.”

Mr. Schumer also said the United States should consider reaching out to allies to try to move the entire summit, which includes leading developed and developing powers, away from St. Petersburg.

“For once, Bob, I agree with Chuck Schumer on that,” Rep. Paul Ryan, Wisconsin Republican, said later on the program. “I think President Putin thinks he can get away with pushing around this administration because the administration has given sort of appeasement feelings that they can do this. The ‘reset’ policy has been an utter failure. This is a stab in the back. This is a slap in the face. … That has to come with consequences.”

Former NSA and CIA Director Gen. Michael V. Hayden said on “Fox News Sunday” that Mr. Obama should not go to the planned meeting with Mr. Putin, and that it was a “jump ball” on whether Mr. Obama should go to St. Petersburg for the G-20 summit at all.

Chairman of the Joint Chiefs of Staff Gen. Martin E. Dempsey said Mr. Snowden is not acting for honorable or noble purposes, or to stir up a debate over security and privacy rights. He said on ABC’s “This Week” that he does not know how much classified information Mr. Snowden has, but that it is “obviously significant.”

“He has caused us some considerable damage to our intelligence architecture,” Gen. Dempsey said. “Our adversaries are changing the way that they communicate. My job is to protect the country. So I am very concerned about this.”

Mr. Snowden’s revelations have sparked a national debate on the surveillance programs and their oversight. Some lawmakers, including Sen. Saxby Chambliss, the ranking member on the Senate Select Committee on Intelligence, defended the NSA surveillance programs Sunday as necessary for protecting national security.

“These programs are controversial. We understand that. They are very sensitive, but they are also very important because they are what allow us to have the ability to gather this chatter that I referred to,” said Mr. Chambliss, Georgia Republican. “If we did not have these programs, then we simply wouldn’t be able to listen in on the bad guys.”

Rep. Adam B. Schiff, California Democrat, appeared more skeptical.

“I think you have to be very careful about how much you represent that any particular program has contributed to our security,” Mr. Schiff said on CNN’s “State of the Union.” “I think that, with respect to any of the NSA programs, we need to ask ourselves three questions: We need to ask whether it’s constitutional, whether it’s effective, and whether it’s structured in a way that minimizes any unnecessary imposition on our privacy.”

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