"I am not going to pay for that fucking wall. He should pay for it. He's got the money," Fox told Fusion in an excerpt of the interview on the US television network's website.
Trump quickly hit back on Twitter, writing: "Vicente Fox horribly used the F word when discussing the wall. He must apologize! If I did that there would be a uproar!"
Asked whether he was afraid that the front-runner for the Republican nomination could be the next US president, Fox said: "Not at all. Democracy cannot take crazy people that doesn't know what's going on in the world today."
"I'd like to know who those hispanics are because they again, they're followers of a false prophet," he said.
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"Please, you Hispanics, Latinos in the US, open your eyes. It's not to defend our race. It's not to defend our creed. It's to defend this very same nation that is hosting you. This nation is going to fail if it goes into the hands of a crazy guy," Fox said.
"Mexican people, we are not going to pay any single cent for such a stupid wall. And it's going to be completely useless," Calderon, who was president from 2006-2012, told CNBC. "The first loser of such a policy would be the United States."
Trump has angered Mexico by saying that the country was sending rapists across the border and has vowed to make the country pay for a wall if he is elected.
President Enrique Pena Nieto's administration has called those comments "prejudiced and absurd."