The 19-year-old played in all three of Ukraine's matches in Euro 2016, coming on as a substitute against Germany and Northern Ireland, and starting against Poland.
The Russian club says a "long-term cooperation" agreement between the two clubs is part of the deal.
Zinchenko is a product of the youth academy at Shakhtar Donetsk from where he joined Ufa.
The midfielder is new City manager Pep Guardiola's second signing after Nolito, who joined from Celta Vigo last week.
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(Reporting by Neil Robinson. Editing by Adrian Warner.)
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