A final vote on the measure is scheduled for Tuesday night. The law would grant full marriage recognition, including property inheritance and ownership, guardianship and adoption within city limits, The Arizona Republic reported.
Bisbee is about 200 miles southeast of Phoenix, on the Mexican border. Its population was listed as 5,633 in 2011.
After the council voted unanimously in favor of the measure in an initial vote, the conservative Center for Arizona Policy, sent a letter to council members, charging they didn't have the legal authority to permit same-sex marriage in the city.
Councilman Gene Conners said the protest would not affect the vote.
"I think there are a couple of votes that might peel off," he said. "But if it was easy, everybody would be doing it."