He becomes the Championship club's ninth summer signing after penning a three-year contract.
Villas-Boas says he found out about Marseille signing the Celtic star from the press on Tuesday.
West Ham are also targeting Cagliari midfielder Nandez.
Olivier Ntcham has never actually played for Man City and yet he has just cost Celtic £4.5m.
We use the inverted quotes because Olivier Ntcham has never actually played for Manchester City.