The 21-year-old has been linked with a big-money move to the Etihad Stadium throughout the summer transfer window, and he is believed to be Pep Guardiola's primary defensive target.
It was reported last month that Man City had reached an agreement to sign the Croatia international, but Leipzig head coach Marco Rose and sporting director Max Eberl have played down suggestions that an agreement is in place.
Negotiations remain ongoing, though, and Gvardiol has informed Leipzig of his desire to join the treble winners this summer.
Leipzig are understood to have set a deadline of August 12 for the Citizens to agree a fee for Gvardiol, who is currently taking part in pre-season training with the German club.
Despite recent reports claiming that Liverpool could attempt to hijack City's move for Gvardiol, the Citizens remain optimistic that they can win the race for the defender's signature this summer.
According to the Daily Mail, Man City are pushing to recruit Gvardiol before Sunday afternoon's Community Shield clash with Arsenal at Wembley Stadium after making significant steps in negotiations over the past 24 hours.
Man City spent their pre-season in both Japan and South Korea, but they made the journey back to the UK last week in preparation for their trip to London to face the Gunners.
The report adds that the Citizens held encouraging talks with Red Bull joint-CEO Oliver Mintzlaff on Tuesday and there appears to be a desire from both sides to find a quick resolution.
Man City have been quoted €100m (£86m) by Leipzig for Gvardiol throughout this summer, but it is
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