Vice President Joseph R. Biden Jr. took the unusual step of telephoning President Rafael Correa of Ecuador to urge him not to give asylum to Mr. Snowden. Senior State Department officials have also pushed Venezuela, one of the three countries offering to shelter him, with both sides keenly aware that hopes for better ties and an exchange of ambassadors after years of tension could be on the line.

And all across the region, American embassies have communicated Washington’s message that letting Mr. Snowden into Latin America, even if he shows up unexpectedly, would have lasting consequences.

“There is not a country in the hemisphere whose government does not understand our position at this point,” a senior State Department official focusing on the matter said recently, adding that helping Mr. Snowden “would put relations in a very bad place for a long time to come.”

“If someone thinks things would go away, it won’t be the case,” the official said.

But Washington is finding that its leverage in Latin America is limited just when it needs it most, a reflection of how a region that was once a broad zone of American power has become increasingly confident in its ability to act independently.

“Our influence in the hemisphere is diminishing,” said Bill Richardson, a former American ambassador to the United Nations who visited Venezuela this year as a representative of the Organization of American States. “It’s important that the Obama administration and Secretary of State Kerry devote more time to the region and buttress our relationship with some of the moderate countries, like Mexico and Colombia and Brazil and Peru, to resist that anti-U.S. movement.”

At the same time, Mr. Richardson said, there should be efforts to build bridges to countries antagonistic to the United States.

The countries offering to take in Mr. Snowden — Venezuela, Nicaragua and Bolivia — belong to a bloc of governments engaged in a constant war of words with the United States. Venezuela and Bolivia have expelled American ambassadors and other officials, and in a television interview this week Venezuela’s foreign minister openly shrugged off the American pressure campaign.

“The State Department and the government of the United States should know that Venezuela learned a long time ago and defeated pressures from any part of the world,” the minister, Elías Jaua, said.

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 the airborne misadventure last week 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.

Bolivian authorities called the episode a hijacking, saying the reason was unfounded suspicions that Mr. Snowden was on board, and they accused the United States of being behind it. They also accused Spain, Portugal and Italy of refusing to allow Mr. Morales’s plane to fly over or land in their countries. Latin American leaders quickly rallied to his side, condemning the treatment as an affront to the entire region.

But for all the bluster, it is possible that no government in the region is really eager to see Mr. Snowden land in its country. None of the countries that have offered him asylum have said they would be willing to go fetch him — a potentially complicated undertaking, given what happened to Mr. Morales’s aircraft.

Mr. Richardson said that he was baffled by the stance of Venezuela’s president, Nicolás Maduro. He met with Mr. Maduro in April, just before he was elected, and said he was asked to tell Washington that Venezuela wanted to improve relations, which have been rocky for years.

Mr. Maduro then sent his foreign minister, Mr. Jaua, to shake hands with Secretary of State John Kerry, and they agreed to start talks that would eventually lead to a new exchange of ambassadors. But it seems clear that any hopes for better relations would be scuttled if Mr. Snowden were given safe haven.

“What I think is going on among Bolivia, Venezuela and Nicaragua and possibly others is, who can replace Chávez as the main U.S. antagonist?” said Mr. Richardson, referring to Venezuela’s former president, Hugo Chávez, who died in March. “But the risk for them is a diminished relationship and possibly some retaliation with the U.S. They may feel the headlines they get from being anti-U.S. is worth it for them domestically.”

Ultimately, Nicaragua would be loath to anger the United States, its principal trading partner, especially as it awaits an annual State Department assessment that helps it get international loans and the expansion of a trade preference that allows some of its products to enter the United States duty-free, said Carlos F. Chamorro, a Nicaraguan analyst critical of the government. He argued that the asylum offer made by Nicaragua’s president, Daniel Ortega, amounted to grandstanding, hedged by a caveat that the offer stood “if the circumstances permit.”

“It’s consistent with Ortega’s policy to provoke up to a certain point the American administration while at the same time doing everything to maintain better relations,” Mr. Chamorro said.

Still, Washington’s push for extradition has poked at a sore spot for several countries that have sought the extradition of people wanted by their justice systems.

Mr. Correa has pointed to the case of two brothers, William and Roberto Isaias, who ran a bank at the center of a huge Ecuadorean financial scandal in the 1990s. They were convicted in absentia of financial wrongdoing in an Ecuadorean court. They now live in the United States, but repeated requests for extradition have been unsuccessful.

And Venezuela has demanded the extradition of Luis Posada Carriles, a former C.I.A. operative accused here of masterminding the bombing of a Cuban airliner that killed 73 people in the 1970s. He escaped from a Venezuelan prison in the 1980s and went to live in the United States.

“The first thing you need to do to have the moral standing to ask for the extradition of this youth Snowden, whose only act is to reveal the crimes that you committed, is to turn over Luis Posada Carriles, who you are protecting,” Mr. Maduro said this month.