Reuters could not confirm the authenticity of the Twitter accounts from which the threats were allegedly made.
Trump, a real estate billionaire who has generated controversy by saying many illegal immigrants from Mexico are criminals and rapists, said in a statement on Sunday that "corrupt Mexican officials" had let Guzman escape.
A Twitter account made out in the kingpin's name, Joaquin Guzman Loera, that on Sunday had celebrated his escape sent a message threatening Trump if he continued to speak out.
"Keep screwing (with us) and I'm going to make you eat your fucking words you lousy white faggot," said the Twitter account with the user name @ElChap0Guzman.
Four Mexican government officials said they could not say whether the account, as well as several others in the name of some of Guzman's children that were linked to it, were genuine.
"I am told they are apocryphal," said Mexican Deputy Interior Minister Roberto Campa.
A Trump aide said in a statement: "The FBI is fully aware of the situation and is actively investigating this threat against Mr. Trump."
It quoted Trump as saying: "I'm fighting for much more than myself. I'm fighting for the future of our country which is being overrun by criminals. You can't be intimidated. This is too important."
An FBI spokesman said the agency had no statement on the issue. "Standard practice is to neither confirm nor deny FBI investigations," he said.
Guzman, who escaped from a maximum-security prison through a tunnel, ran the powerful Sinaloa cartel after an earlier prison breakout in 2001 until he was arrested last year.
The cartel has smuggled billions of dollars worth of cocaine, marijuana and methamphetamines into the United States.
White House spokesman Josh Earnest said Guzman posed a threat to public safety in the United States, and that Washington has "made quite clear to the Mexicans our interest in ensuring that he faces justice here in the United States."
In his statement on Sunday, Trump said: "Corrupt Mexican officials obviously let him go a second time. The last time he was free for 13 years. He has been selling drugs in the U.S. big time - a major kingpin."