The Daily Mail says Wirtz's parents, Hans and Karin, were in Manchester yesterday for talks with City directors.
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Wirtz is also a target for Bayern Munich and is said to have reduced his choice of destinations between the two giants.
It's been suggested Hans favours a move to Bayern, while Karin believes he should stay in Leverkusen for another year.
At City, they see Wirtz as a natural successor for departing legend Kevin de Bruyne.
Wirtz has 16 goals and 15 assists in 44 matches for Leverkusen this season.