The 21-year-old midfielder, whose first year at Stamford Bridge was disrupted by serious injury, was confirmed as a target for the Italian giant on Wednesday.
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Van Ginkel is understood to have joined in a season-long loan deal, though the clubs are yet to announce the specifics.
The youngster follows Fernando Torres in swapping Jose Mourinho’s side for Milan, which began its Serie A campaign by beating Lazio 3-1 on Sunday.
The Spain international striker has agreed a two-year loan with Filippo Inzaghi's side after consistently disappointing at Stamford Bridge since his €63 million move from Liverpool.