Bobby and Cathy Horvath say the order will force them to relocate their Wildlife in Need of Rescue and Rehabilitation from Oyster Bay.
An Oyster Bay spokeswoman told Newday neighbors have complained to the town about the property being dirty and smelly.
Town officials are working with the Horvaths on a flexible timetable.
Town Supervisor John Venditto says he supports the couple's work. But he says neighbors don't feel "comfortable having a bobcat within striking distance."
Last week, the 6,000-square-foot property was housing seven hawks, two turkeys and the bobcat.
The Horvaths say they have federal and state permits for their operation.