Steve Martan, a campus monitor at Miles Exploratory Learning Center in Tucson, left three voice mails for Congresswoman Martha McSally, the FBI said.
McSally, a Republican, represents the Second Congressional District in the US House of Representatives, which includes the city of Tucson.
In the voicemails, which were made on May 2 and May 10, Martan threatened to shoot the congresswoman between the eyes, the FBI said.
According to a criminal complaint, Martan told McSally she needed to "get back to where [she] came from and leave Arizona" and warned that she should "be careful" when she returned to Tucson.
The FBI said agents tracked down Martan by his phone number through his cellphone provider and arrested him May 11.
Upon his arrest, Martan admitted to calling McSally's office, the FBI said. He told agents he "was venting frustrations with Congresswoman's McSally's congressional votes in support of the President of the United States," the complaint said.
A 'wake-up call'
A statement from McSally's office said the threats crossed "a clear line" and were "a wake-up call."
"Threatening to shoot a member of Congress between the eyes and stating that her days are numbered is sickening. It is especially sickening here in southeastern Arizona because we know, perhaps better than any congressional district in the country, what happens when threats of violence become acts of violence," staffer CJ Karamargin said in the statement.
"The January 2011 shooting of Congresswoman Gabrielle Giffords was followed by a national discussion about the importance of civility and respectfulness in our public debates."
Giffords and 18 others were shot by a lone gunman outside a supermarket
in her Tucson-area district. Six of those victims died, but Giffords, who was shot in the head, made a miraculous recovery.
Walter Goncalves Jr. is a public defender who is representing Martan. He told CNN he can't comment on the complaint. He says Martan's next court appearance will likely be a change of plea hearing or some pre-trial hearing. Martan's arraignment will be in approximately 30 days.
Martan was released May 12 and is on house arrest. He must attend all future proceedings, stay away from McSally, and avoid drugs and alcohol. He is also not allowed to possess any firearms and must participate in a mental health treatment program. The U.S. Attorney's Office of Arizona and the FBI had no comment.