"We would like for him to pay first for his crimes here," Mexican Interior Minister Miguel Angel Osorio Chong told reporters.
Osorio said the United States had not made an official extradition request for Guzman, who was arrested Saturday by Mexican marines acting on U.S. intelligence, but would take it under advisement if one were made, the Los Angeles Times reported.
The minister said Guzman had not been offered status as a protected witness by U.S. or Mexican authorities, the newspaper said.
The Mexican attorney general's office forensics division chief told reporter handwriting samples, fingerprints and DNA tests were used to confirm the suspect arrested last weekend is, in fact, Guzman.
"We scraped the inside of his mouth, got the genetic profile ... and it was a positive match," Sara Monica Medina said.
The Times said a Strategic Cabinet polling agency poll found 42.2 percent of Mexicans had doubted the suspect was Guzman and 40.7 percent flat out didn't believe it.
And 68.8 percent said they believed he could escape just as he did in 2001. He originally was taken into custody in the 1990s.
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