“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.