A US judge has admonished the lawyer for drug smuggler Joaquín “El Chapo” Guzmán over his opening statement in which he accused two Mexican presidents of taking bribes.

Federal prosecutors had asked US district judge Brian Cogan to throw out the defense’s opening statement at Guzmán’s New York trial, saying it was “permeated with improper argument, unnoticed affirmative defenses and inadmissible hearsay”.

Cogan stopped short of that on Wednesday, but admonished defense attorney Jeffrey Lichtman for having gone “far afield of direct or circumstantial proof”. He said he would instruct the jury to focus on the evidence.

“Your opening statement handed out a promissory note that your case is not going to cash,” the judge said, calling Lichtman’s opening misleading.

In his opening statement on Tuesday, Lichtman told jurors that US prosecutors had dreamed “for decades” of convicting Guzmán, adding: “The world is focusing on this mythical El Chapo figure.”

Lichtman told the court his client was not the real leader of a cartel that sent tons of cocaine into the US, and sought to shift blame to Ismael “El Mayo” Zambada, another alleged drug trafficker in the cartel’s leadership.

Zambada remains on the loose in Mexico, Lichtman claimed, because of bribes that “go up to the very top”, including hundreds of millions of dollars paid to the current and former presidents of Mexico.

“This is a case that will require you to throw out much of what you were taught to believe in about the way governments work and how they behave, governments in South and Central America and Mexico and even the United States,” Lichtman told the jurors. “This is a case which will require you to open your minds to the possibility that government officials at the very highest level can be bribed, can conspire to commit horrible crimes – that American law enforcement agents can also be crooked.”

A spokesman for the Mexican president, Enrique Peña Nieto, called the allegations “completely false and defamatory”. Former president Felipe Calderón dismissed them as “absolutely false and reckless”.

Guzmán, who has been held in solitary confinement since his extradition to the United States early last year, has pleaded not guilty to charges that he amassed a multibillion-dollar fortune smuggling tons of cocaine and other drugs in a vast supply chain that reached well north of the border.

Lichtman resumed his opening statement on Wednesday, describing “the myth of El Chapo” as “very strong”. He said law enforcement agents had his client autograph $100 bills for them upon his arrest.

“Mr Guzmán was somebody who enjoyed the publicity,” Lichtman told the jury. “He enjoyed the notoriety.”

The defense attorney also described the government’s witnesses as liars seeking to mitigate their own jail sentences.

“These are people who have lied every day,” he said. “They’re here because they want to get out of jail by any means necessary.”