Last March, Mr. Trump also openly floated the idea of the designation, saying in an interview with Breitbart News that he was thinking “very seriously” about it.

But a groundswell of support, mainly among conservatives, gained momentum in recent weeks after the killing of three mothers and six of their children, all dual Mexican and American citizens, who were part of a fundamentalist Mormon community in northern Mexico.

The State Department declined to comment on Wednesday, referring several questions to the White House about how many cartels Mr. Trump was looking to designate, or when a decision might be made.

It is not new for the United States to consider listing drug cartels as terrorist organizations. Members of Congress have made similar requests during past presidential administrations, said Jason Blazakis, the former director of the State Department’s office of counterterror finance and designations.

But, Mr. Blazakis said, the United States has largely resisted adding them to official terror organization lists out of concern that it would be a mostly symbolic step that eats up already-scarce resources and could do more damage to diplomacy than good.

Other analysts warned that the designation could undermine or disrupt the carefully tuned relationship between the United States and Mexico, a complex arrangement developed over generations of administrations and covering a broad range of issues, including trade, crime-fighting, finance, migration, tourism and culture.

Arturo Sarukhán, a former Mexican ambassador to Washington, warned that the Trump administration could use the terrorist designation to block loans to Mexico from international financial institutions and even restrict imports to the United States from Mexico.