Barton claimed that the Scottish Premiership champions made a move for him before he completed his switch to Glasgow Rangers.
Brendan Rodgers has been appointed the new Celtic manager.
Celtic chief executive Peter Lawwell has hailed new manager Brendan Rodgers
Celtic are looking to bring in a new manager to succeed Ronny Deila.
Former Bhoys David Moyes and Roy Keane may be put off Parkhead returns, according to Sutton.
The Hoops chief has been linked with a job with the Premier League side.
Celtic chief executive Peter Lawwell speaks highly of Scott Brown.
The Norwegian will leave the Scottish Premiership side at the end of the season.
Ronny Deila will leave the Scottish Premiership side at the end of the season.
Peter Lawwell claims Kieran Tierney wants to stay at Celtic.