South Yorkshire Operational Support Unit tweeted pictures of officers on motorbikes patrolling the course and damage to one green.
The tweet said: “Not the usual patrol route, alongside the 8th hole on Concord Park Golf Course. But some donuts really left their mark here. Over half a dozen greens trashed! We will find you.”
The golf course is part of Concord Park in Shiregreen, next to the sports centre.
We have contacted the club for comment.