The 19-year-old has just six months to run on his current deal and could leave for a cut-price fee at the end of the season if an extension is not agreed.
City acquired Dilrosun from Ajax in 2014, beating off competition from a host of sides to sign the winger. Those same teams remain interested in the Netherlands youth international, with Liverpool said to be one of them, according to ESPN.
Serie A holders and last season's Champions League runners-up Juventus, Bundesliga pair Borussia Dortmund and RB Leipzig and La Liga side Valencia are understood to be the other sides keeping an eye on the situation.
Dilrosun is yet to make his first team debut for City but has represented their Under-18, Under-19 and Under-23 sides as well as turning out for the club in the Uefa Youth League. In the summer, he was a key part of the Netherlands squad which reached the semi-finals of the Under-19 European Championships.
Pep Guardiola has not been shy in rewarded youngsters with senior appearances, with the likes of Brahim Diaz, Oleksandr Zinchenko and Phil Foden all making their debuts this season.
But with Dilrosun seemingly no closer to being given a chance, Liverpool hope they can tempt the teenager into moving to Merseyside by offering him guaranteed route into the first team.
Jurgen Klopp, the Liverpool manager, has entrusted youth throughout his tenure at Anfield with the likes of Ben Woodburn and Trent Alexander-Arnold the latest examples of the German's faith in the club's academy.
Both players are now regular fixtures in the Liverpool first team squad, with Woodburn even going on to earn international recognition with Wales earlier this season.
However, it remains to be seen whether a promise to provide Dilrosun with direct passage into the first team will be enough to tempt him away from City where he has been since he was 16.
City can expect to recoup a fee for Dilrosun, who has not reached the age whereby he is eligible to depart on a free, but talks with a rival club are likely to end in a tribunal to determine the price.