Benfica also said that City defender Nicolas Otamendi will go the other way in a separate transfer for €15m.
Benfica confirmed the transfers in a statement on Sunday but said the players would still need to sign their respective contracts.
Dias, 23, has made 137 appearances during a four-year spell with Benfica, winning the league title in 2018-19.
He has won 19 caps for Portugal and was part of the side that lifted the UEFA Nations League trophy last year.
His arrival will help shore up a leaky City defence which shipped five goals to Leicester City at home on Sunday.
Otamendi, 32, won two Premier League titles, four League Cups and an FA Cup during his five-year spell at the Etihad.