The Porto centre-back has been on both Manchester City and Chelsea's radar this season, but the report claims Manuel Pellegrini has made the first move for the French international defender.
Big-spending Ligue 1 side Monaco have also been keeping tabs on the 22-year-old, who made his debut for France earlier this year after 23 appearances for the under-21 national team.
The report claims the €45m bid is the same amount City offered Porto at the end of the summer transfer window - a proposal the Portuguese club turned down.
But it could be different this month with the player thought to be keen to move to the Premier League.
He may wait until the end of the season to enjoy a full preseason with his prospective new club, but the report suggests City want him in before the end of January.
The bid may prompt Chelsea into action, as the Stamford Bridge club have been strongly linked with Mangala.