Barnett joined Peterborough for £1.1million last summer following a loan spell from Crawley Town, but he is currently on the transfer list at London Road.
Millwall made an enquiry for the 27-year-old last month and Posh chairman Darragh MacAnthony confirmed on Twitter earlier this week that Blackpool had also shown an interest.
Peterborough's director of football Barry Fry has been fielding calls from an interested Championship club and he expects Barnett to move on.
"I am close to selling Tyrone," Fry told the Peterborough Evening Telegraph. "A Championship club are very, very interested judging by the number of times they have contacted me.
"We expect to get over a million for him which would be great business for the club."
Barnett rejected the opportunity to join Blackpool from Crawley in January 2012 before his move to Peterborough, but the Seasiders are now under the management of Paul Ince.
He has scored five goals in 31 appearances in total for Peterborough and also spent a brief spell on loan at Ipswich Town last season.