Hoops manager Ronny Deila revealed last month that Forrest has snubbed a new contract at Parkhead, and barring a u-turn, the Scotland international will be available on a Bosman free transfer later this year.
But while the Mirror claim his contract expires in the summer, others - including Sky Sports - report that he is tied to the Scottish champions until the end of December.
Nevertheless, Norwich manager Alex Neil has received the go-ahead to strengthen his squad once the transfer window reopens, according to the Mirror, and plans to add Forrest - against whom he played during his Hamilton days.
Given Sunderland's improvement under Sam Allardyce, Norwich are facing an uphill battle for survival, and unless any additional interest materialises, Forrest looks to be Championship-bound next term.
Both QPR and Forest have struggled in the second tier this season and sit 12th and 15th respectively, despite their pre-season promotion aspirations.
Forrest, 24, has played 188 times for Celtic, scoring 30 goals, and has 13 caps for Scotland.