A post riddled with obscenities on a Twitter account, reportedly run by the drug lord’s son Ivan, said the billionaire businessman would be made to “swallow his words”.
That was immediately followed by a post suggesting someone would be killed, although it was not clear if that referred to Mr Trump. Associates who had betrayed Guzman were also targeted.
He posted a message to Mexican president Enrique Peña Nieto saying: ”And you @EPN, don’t call me a delinquent because I give people work unlike you you cowardly politician.’
It could not be determined who had made the comments but the account went on to proclaim that “no cage could hold” El Chapo.
On a separate account linked to Ivan Guzman he had appeared to foretell his father’s escape a week ago, saying:
“Good things come to those who wait.”
Guzman, who is believed to be 58, was widely considered the world’s richest and most powerful drug trafficker before his capture, last year, and cultivated a Robin Hood image among his followers as he ruled over the Sinaloa cartel.
After he escaped guards at Altiplano prison, 55 miles west of Mexico City, entered his cell and found a hole 10 metres deep with a ladder.
It led to a tunnel a mile long and 1.7 metres high, which allowed the diminutive drug lord to walk upright to freedom.
Mr Trump had seized on the prison escape to back up his calls for a wall between the two countries, and said it proved the extent of corruption in Mexico.
He added: “Can you envision Jeb Bush or Hillary Clinton negotiating with El Chapo…Trump, however, would kick his ass!”
The entrepreneur’s poll numbers have been rising since last month when he began controversially attacking Mexican illegal immigrants as the cause of problems in the US.
He appeared to be unfazed after being targeted by Guzman and his supporters on social media.
Mr Trump responded: “El Chapo and the Mexican drug cartels use the border unimpeded like it was a vacuum cleaner, sucking drugs and death right into the US. We get the killers, drugs and crime, they get the money.”
But Mr Trump said he had contacted the FBI to investigate the apparent threat.
The property mogul told TMZ: “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. It’s too important.”
The escape was a major embarrassment for Mr Pena Nieto. Speaking on a visit to France, he called it “an affront to the state”.
Guatemala launched an operation to secure its border, which Guzman fled across once before.
Mexican prosecutors began questioning 30 prison employees, including the warden, amid suspicions of an inside job.
Mike Vigil, a retired US Drug Enforcement Administration operations chief, said if Guzman was not captured with two days of his escape he may never be found.”
“If he is able to make his way to Sinaloa, his native state, and gets into that mountainous range, it’s going to be very difficult to capture him because he enjoys the protection of local villagers.”
Raul Benitez Manaut, a security expert at the National Autonomous University of Mexico, said: “El Chapo surely planned this from the time he was jailed and had very large internal and external support to escape.
“There certainly was corruption inside and outside the prison. It was a film-like escape.”