Coppinger, 31, who has been with Doncaster since 2001, has been the subject of speculation this summer with Rovers in the process of off-loading several senior top-earning players to cut their wage bill following relegation from the Championship.
The likes of Brian Stock and James O'Connor have already left the Keepmoat Stadium.
Former Doncaster boss Sean O'Driscoll, now at Nottingham Forest, is a known admirer of Coppinger, while Hull and Ipswich have also recently been linked with the player.
South Yorkshire rivals Barnsley have now thrown their hat into the ring by contacting Rovers.
Tykes boss Keith Hill told the Yorkshire Post: "We've made an enquiry about James and we'll have to see what happens. Hopefully, we can do something by the weekend."