Canaries boss Chris Hughton revealed over the weekend he was prepared to let Barnett leave Carrow Road this summer.
The 27-year-old still has a year to run on his contract at Norwich, but he does not figure in Hughton's long-term plans.
The news has alerted a host of clubs to Barnett's availability with the defender impressing during a successful loan spell at Cardiff last season.
Doncaster are keen to lure Barnett to the Keepmoat Stadium as Paul Dickov looks to assemble a squad capable of competing in the Championship next season.
A number of other Premier League and Championship clubs are also thought to be weighing up moves for the former West Brom man.