The Germany international is now in self-isolation as the club prepare to play Wolves at Molineux.
"Ilkay is now observing a 10-day period of self-isolation in accordance with Premier League and UK government protocol on quarantine," said a City statement.
"Everyone at the club wishes Ilkay a speedy recovery."
Gundogan is the third City player to return a positive test this month, following Aymeric Laporte and Riyad Mahrez.
City missed the opening weekend of the Premier League to give them extra time to recover from a late end to the 2019/20 season due to their involvement in the Champions League.
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