The 18-year-old England U19s winger has a year left on his current deal and the Whites, who are determined to keep him, have offered Roberts a new long-term contract.
Sky sources understand City are very interested in the attacker after the Premier League’s second-placed finishers were impressed with his performances in the recent U19 European Championships.
Roberts scored four goals and provided four assists in the qualifying rounds in October before finding the net again in the elite round defeat to France in March – a result which confirmed England’s exit from the tournament.
Roberts was thrust into Fulham’s first team by former manager Felix Magath during their relegation season from the Premier League, making his debut against Manchester City in March 2014.
Magath’s successor Kit Symons continued to play the attacker sparingly in the Championship last season, with Roberts making two starts in his 17 league appearances.
He joined the Whites from AFC Wimbledon at the age of 13 before signing his first professional terms at 17 last February.