The paper claim Neil Lennon's side are keen on a season long loan deal for the winger.
HITC reported earlier this month that Leeds United were keen on Barker, as they consider their options to solve a problem position in their side.
Barker has impressed on Manchester City's pre-season tour, even scoring a goal for the first team.
The teenager is however in with a tough challenge to nail down a first team place this coming season, not helped by the big money arrival of Raheem Sterling.
He netted four goals and provided four assists in the UEFA Youth League last season, a competition ultimately won by Chelsea's under-19 side.
Bolton did well out of loan deals last season, bringing in Saidy Janko from Manchester United and Adam Le Fondre from Cardiff City.
They are currently working on a limited budget, but even so the arrival of Barker would be quite a coup even if they did have plenty of money to spend.
Bolton and Leeds are unlikely to be the only sides interested in his signature, but despite the competition for places he has at City at present, the club are yet to indicate whether they prefer to send him out on loan or keep him at close hand to try and give him first team opportunities, albeit limited.