The man, Ronald Lee Syvrud, 66, had multiple outstanding warrants from Wisconsin, the sheriff's office said. He was taken into custody on Thursday afternoon, Cochise County Sheriff's Office said on Friday.
Trump, the Republican nominee for president, was in Cochise County on Thursday when he visited the US border with Mexico as part of his campaign.
US presidential candidate Donald Trump says he isn't surprised by the latest threat on his life. (AP PHOTO)
Syvrud was booked into the Cochise County Jail on a felony warrant from Graham County in Arizona for "failure to register as a sex offender and two counts of threatening for this case," the sheriff's office said.
He remained in custody pending further court adjudication.
"I am not that surprised (by the threat) and the reason is because I want to do things that are very bad for the bad guys," Trump said on Thursday when asked about that threat against him in Arizona.
It was the latest in a series of threats against candidates ahead of the November 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, an attempted assassination on Trump left two others injured and one man dead.
The US Secret Service came under widespread scrutiny following that attempt. It resulted in the resignation of the agency's director.
Despite this and new threats, Trump on Thursday said he has "great respect" for the Secret Service and "the job they do", while adding they made mistakes from which they will learn.
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