The Gulf Arab nation Qatar is trying to act as a mediator in the conflict between the United States and Venezuela, even as President Trump continues building up military forces in the Caribbean and striking civilian boats, according to three people with knowledge of Qatar’s diplomacy.

Qatar’s efforts have been encouraged by the Venezuelan government led by President Nicolás Maduro, but they have not been embraced by the Trump administration, which appears more focused on military options than on diplomacy.

The Pentagon has deployed 10,000 U.S. troops to the region, most of them to bases in Puerto Rico, a senior U.S. military official said. Troops are also on eight surface warships and a submarine in the region.

Marco Rubio, the secretary of state and national security adviser, says Mr. Maduro is an illegitimate leader and a fugitive from a 2020 Justice Department indictment on drug trafficking charges. Mr. Rubio has been trying to craft a strategy to oust Mr. Maduro through military pressure, a mission supported by John Ratcliffe, the C.I.A. director, and Stephen Miller, Mr. Trump’s chief domestic policy adviser.

“I do know Qatar is passing messages back and forth,” said Juan Gonzalez, who was the director for Western Hemisphere affairs on the National Security Council during the Biden administration. “They’re trying to find a way to encourage a more structured dialogue or back channel between both sides, but they have not been getting much traction with the Trump administration.”

A current official described Qatar’s efforts as trying to keep channels of communication open between the United States and Venezuela as part of the tiny nation’s goal of playing an important role in global diplomacy.