The future of Arlington County’s governing system has been thrown into doubt as the county’s election staff tallies up signatures on a petition that would put a change-of-government referendum on the ballot this fall.

The petition requires signatures from 10 percent — or 14,350 — of the county’s more than 143,000 registered voters. The county’s voter registration office on Monday had verified nearly 9,000 of the more than 16,400 petition signatures submitted last week.