Donald Trump, the Republican nominee for president, is in Cochise County as he visits the U.S. border with Mexico as part of his campaign.
The man, Ronald Lee Syvrud, 66, has multiple outstanding warrants from Wisconsin, the sheriff's office said.
"No, I have not heard that but I am not that surprised and the reason is because I want to do things that are very bad for the bad guys," Trump said when asked about that threat against him in Arizona.
This is the latest in a series of threats against candidates ahead of the Nov. 5 general election.
In early August, a Virginia man was charged with threatening Vice President Kamala Harris, Trump's Democratic rival, and a New Hampshire man was arrested in December for threatening to kill Republican presidential candidates.
In July, Trump was shot in his ear in an attempted assassination that left two others injured and one man dead. The U.S. Secret Service came under widespread scrutiny following the shooting. It resulted in the resignation of the agency's director.
Despite this and new threats, Trump said he has "great respect" for the Secret Service and "the job they do," while adding they made mistakes from which they will learn.