City boss Pep Guardiola is a big admirer of the 21-year-old centre-back, who starred at the World Cup last year.
BBC Sport has been told contact has been made between City and Gvardiol’s camp.
Gvardiol has a release clause in excess of 110m euros (£94.35m), which can be triggered next summer.
It means Leipzig are not willing to let him leave 12 months early for a significantly smaller amount.
Gvardiol made 41 appearances for Leipzig last season and scored against City in the first leg of their Champions League last-16 tie, which they went on to lose 8-1 on aggregate.
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