Baleba, 22, becomes Manchester United's third major midfield signing of the transfer window, following the arrivals of Youri Tielemans from Aston Villa and Andrey Santos from Chelsea.
According to the BBC, United will pay £65 million initially, with a further potential £5 million at a later stage.
Baleba has signed a five-year contract with an option for another year.
Manchester United had a miserable start to the Premier League season and lost 2-0 away to Swedish World Cup defender Elliot Stroud's new club, Hull.
Manchester United have spent £153 million, equivalent to almost two billion kronor, on signings during the summer.
