WASHINGTON—The Trump administration’s attempt to force out the president of Venezuela marked the opening of a new strategy to exert greater U.S. influence over Latin America, according to administration officials.

In sight isn’t just Venezuela’s Nicolás Maduro, but also Cuba, an antagonist that has dominated American attention in the region for more than 50 years, as well as recent inroads made by Russia, China and Iran.

While Mr. Maduro and his predecessor, Hugo Chávez, have long drawn Washington’s condemnation, the Trump administration is stocked with officials who have long believed Cuba to be the more serious national-security threat. They cite Cuba’s intelligence operations in the U.S., and its efforts to spread anti-American views in other Latin American countries.

The goal, the administration’s thinking goes, is to sever ties that bind Venezuela to Cuba and sink regimes in both countries.

The emerging U.S. assertiveness stems from the desire of the White House to reverse a partial rapprochement with Havana by the Obama administration through the easing of sanctions and the island’s opening to U.S. investment.