Maricopa County Chief Deputy Jerry Sheridan made the comment Friday at a news conference in Phoenix.
The package intercepted in Flagstaff late Thursday was addressed to Arpaio at his downtown Phoenix office. It had been left in a parcel locker in a rural part of Coconino County.
U.S. Postal Inspection Service spokesman Keith Moore said a courier brought it into the main Post Office in Flagstaff, and authorities used a water cannon to neutralize the package.
Arpaio, the self-proclaimed "toughest sheriff in America," said this isn't the first time he's been threatened. Arpaio is known nationally for his strict treatment of jail inmates and cracking down on illegal immigration.
"Of course you worry. I'm a victim, I'm a witness. When you convict people, the victim has to be somewhat concerned. I'm a little concerned about my family," Arpaio said at the news conference. "I didn't ask for all these threats."
The package intercepted in Flagstaff late Thursday was addressed to Arpaio at his downtown Phoenix office, the Maricopa County Sheriff's Office said. It appeared suspicious, so it was X-rayed, and an improvised device was detected. A bomb squad team neutralized the device with a water cannon, said U.S. Postal Inspection Service spokesman Keith Moore.
Postal Inspector Patricia Armstrong said authorities were alerted by a "very astute" carrier who observed "something suspicious" about the package when the carrier emptied a parcel locker in a rural part of Coconino County, outside Flagstaff city limits.
Armstrong didn't elaborate on what raised suspicion, but Tom Mangan, a spokesman in Phoenix for the federal Bureau of Alcohol, Tobacco, Firearms and Explosives, said initial reports indicated that the package was a box that might have been damaged in transit and leaked gunpowder.
Authorities in Flagstaff, which is about 140 miles north of Phoenix, said they are pursuing leads in the case.
Arpaio said he receives lots of packages, including ones with cookies recently as he recuperated from a broken shoulder suffered when he fell crossing a street on his way to lunch. He said he opens some of the packages personally.
"Hey, they send me cookies, now they send me bombs," he said Friday. "It's a little different."
Arpaio said the mailing of an explosive device addressed to him comes with his line of work. He cited the recent killings of a West Virginia sheriff, Colorado's corrections director and two prosecutors in Texas.
"That's the nature of the business," he said.
Arpaio said whoever is responsible for mailing the package, if found, would be brought to justice.
"I'm not going to be intimidated by anyone, that's a promise," he said.
Following the killing of a West Virginia sheriff last week, Arpaio said elected law enforcement officials across the nation seem to be targeted.
Numerous threats against Arpaio, a hero to many conservatives on immigration, prompted the need for a security detail for the lawman also known for dressing jail inmates in pink underwear and making them sleep in tents in the heat of the Arizona desert.
A campaign to recall Arpaio began just weeks after he started his sixth term in January.
Critics contend that Arpaio should be ousted because his office failed to adequately investigate more than 400 sex-crimes cases, allegedly racially profiled Latinos in its trademark immigration patrols and has cost the county $25 million in legal settlements over treatment in county jails.
Arpaio has denied that his deputies racially profiled Latinos in traffic patrols targeting illegal immigration. His office has moved to clear up the sex-crime cases and moved to prevent the problem from happening again, he said.
Fonseca reported from Flagstaff, Ariz. Associated Press Writer Paul Davenport in Phoenix also contributed to this report.