The United States has continued to reach out to Venezuela. Roberta S. Jacobson, assistant secretary of state for Western Hemisphere affairs, repeated the Obama administration’s position on Mr. Snowden this week in a phone call with the chargé d'áffaires of the Venezuelan Embassy in Washington, a government official said.

In some cases, the diplomatic effort seems to have paid off. Ecuador at one point appeared eager to grant Mr. Snowden refuge, but it gradually seemed to back off, saying that it could not even consider his request for asylum unless he was in the country or in one of its embassies abroad.

The call from Mr. Biden brought an uncharacteristically warm response from Mr. Correa, who often rails against what he sees as excessive American influence in the region. In an interview, he praised Mr. Biden as being cordial, saying the vice president asked him not to grant asylum and explained that “it could greatly deteriorate relations, but without any kind of threat, just presenting the importance that the Snowden case has for them.”

By contrast, Mr. Correa bristled at what he viewed as threats by American senators who vowed to end trade preferences on some Ecuadorean goods if his country sheltered Mr. Snowden. One group of preferences expires at the end of the month unless renewed by Congress, but Ecuador has sought separate White House approval for duty-free treatment for roses, broccoli and artichokes. The White House said last week it was postponing a decision.

Mr. Snowden’s leaks sometimes appear timed to coincide with where he is at the moment or hopes to go. When he was hiding out in Hong Kong, he leaked documents about American spying in China.

Now it is Latin America’s turn. This week, a Brazilian newspaper, O Globo, has printed articles based on his leaks about how the United States has been collecting data on telephone calls and e-mail traffic in Brazil and other Latin American countries, pushing even close allies of the United States to lodge angry protests with Washington.

The intensity in the region has been fueled in part by last week’s airborne misadventure of President Evo Morales of Bolivia, whose plane was turned back from French airspace and forced to make an emergency landing in Vienna after a meeting in Moscow, where Mr. Snowden has been holed up in an airport.