The 30-year-old, who spent last year on loan at Ligue 1 side Marseille, has started in all four games in the Championship this season, with the Hoops still unbeaten.
Despite being on the lookout for a new club, the once-capped England international could not be lured by the prospect of Premier League football at Selhurst Park by manager Ian Holloway.
When asked about a move to Selhurst Park, Barton replied: "They tried but it's not the right move for me and my family right now.
"Hope they stay up as I like Holloway."
West Ham United are also believed to be considering an offer.