The England under-19 winger is wanted by head coach Uwe Rosler, who has closely watched Barker come through the academy ranks at his former club.
Barker is away with the City first-team squad on their pre-season tour of Australia and came on as a second-half substitute in Tuesday's penalty shoot-out victory over Roma.
City boss Manuel Pellegrini has briefly spoken with the 18-year-old, who has been compared in playing style to Bayern Munich star Arjen Robben.
City want a little more time to review all the options before making a final decision and Leeds may now have to get in line due to the mounting interest.
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