Reports in Italy suggested that the Celtic midfielder was being monitored closely by the Serie A giants.
However, we understand links touting a £7m move are wide of the mark and nothing more than speculation.
The 24-year-old is out of contract at Celtic Park next summer. He joined the club in 2020 from Motherwell.
The midfielder is believed to be considering his options given his contractual situation and status within Brendan Rodgers’ squad.
The Scotland international has made 14 appearances so far this campaign. Seven of those have come from the start, and he has just as many goals.
Renowned Italian scout and intermediary Michele Fratine made the claims with regard to the links between Turnbull and Milan.
Meanwhile, Callum McGregor has urged the Celtic board to ‘help’ manager Brendan Rodgers in January as he tries to build a team capable of competing in European football.
Rodgers has consistently spoken of the need for quality additions to his Celtic squad since the summer, and he reiterated the point after watching his team crash out of European competition for the season with their 2-0 defeat to Lazio in the Stadio Olimpico on Tuesday evening.
Celtic went into the game without the likes of Reo Hatate, Daizen Maeda and Luis Palma, and their back-up cast proved unable to break the team’s wretched run at the top level.
Now, Celtic captain McGregor has backed Rodgers’ call for reinforcements, and he says he is sure that the club’s board will get behind his manager in January.
“We always need help,” McGregor said. “If you ask any manager how many new players does he want, he will always say it’s as many as he can.
“And it’s the same for the players. We want as many good players coming through the door as we can. That builds competition in training and games all the time. And the level just accelerates from there.”
