Learning of their rejection, Mr. Lopes wrote an email to Mr. Menendez.

“Would you rather wait for the outcome of the follow up letter or call the ambo ASAP?” he wrote, using the shorthand for “ambassador.”

“Call ambassador ASAP,” Mr. Menendez replied.

The prosecution asked if it was an unusual step to reach out to an ambassador about a visa inquiry.

“Only if a United States senator advocates on their behalf,” he replied. “It was not uncommon for Senator Menendez to advocate forcefully.”

The defense argued that Mr. Menendez regularly involved himself in visa applications. They cited a case in 2008 involving a family from Colombia who were also initially denied a visa to travel to the United States and showed a letter written from Mr. Menendez to the ambassador advocating on their behalf.

Raymond Brown, one of Mr. Menendez’s lawyers, also sought to dispel any notion that Mr. Menendez’s advocacy of visas, whether for Mr. Melgen’s friends or anyone else, was an abuse of power.

“The language ‘careful consideration,’” Mr. Brown said, quoting words used by Mr. Menendez in the letters he wrote on behalf of both Mr. Melgen’s friends and the Colombian family’s visa applications. “Is that secret code for let her in, or is that exactly what he meant?”

Mr. Lopes answered that it was standard language as part of his advocacy in visa issues.

Throughout Monday’s proceedings, the jury was routinely dismissed as Judge William Walls debated with the prosecution and the defense about rules for permissible evidence, with some arguments extending more than 20 minutes before the jury was brought back in.