Sheriff’s officials say an arrest warrant was issued Monday for 47-year-old Robert Paul Menard of Fall River.
They say deputies have traveled to the Boston area to track down Menard, who had not been arrested as of Wednesday night.
A sheriff’s detective says Menard was identified after allegedly leaving a voicemail on the office’s telephone system recently.
Menard allegedly claimed a bomb was planted at Arpaio’s office in downtown Phoenix, but a search turned up nothing.
The 81-year-old Arpaio started his sixth consecutive term as sheriff of Arizona’s most populous county in January.
America’s self-proclaimed toughest sheriff is known for jailing inmates in tents and making prisoners wear pink underwear.