Twenty-seven-year-old Barnett has been placed on the transfer list at London Road following allegations that he breached club rules.
Speculation last January claimed that Ipswich tried to beat Peterborough to the striker's signature when it was confirmed that he was going to leave Crawley Town.
However, it now appears that the Tractor Boys have finally landed their target, according to East Anglian Daily Times. It is believed that Ipswich are hoping to conclude the deal before the emergency loan window closes on Thursday afternoon.
Barnett has scored just once in 10 appearances for Posh this season.