The Portuguese side announced their centre-half is set to join the Premier League side in a big-money transfer
City defender Nicolas Otamendi will head the opposite way in a separate deal, with both players still needing to finalise their moves by signing contracts with their new teams.
Portugal international Dias, 23, has been a regular for the Primeira Liga side since emerging through the academy ahead of the 2017-18 campaign.
Dias has been linked with a number of big-name clubs around Europe, including Manchester United, but rivals City have now reached an agreement with Benfica.
The Portuguese side confirmed the news on their official website on Sunday, shortly before announcing a £14m (€15m/$17m) deal is in place for Otamendi.
Dias will be a much-welcomed addition to Pep Guardiola’s squad after their defensive weaknesses were laid bare in a 5-2 drubbing by Leicester City at the Etihad on Sunday.
Starting with new signing Nathan Ake and the inexperienced Eric Garcia at centre-half, City were carved open repeatedly by a Foxes side who only had 29 per cent of possession during the 90 minutes.
Golden Boot winner Jamie Vardy snared a hat-trick, including two penalties, while James Maddison and Youri Tielemans finished off the rout in the latter stages.
Manchester City have spent the off-season looking for defensive reinforcements after falling well short in their goal to retain the Premier League title, while crashing out of the Champions League to Lyon at the quarter-final stage.
Pep Guardiola’s side were forced to use veteran midfielder Fernandinho in a central defensive role with Aymeric Laporte spending much of the campaign injured, and Otamendi and John Stones struggling for form.
City were linked with a number of defenders during the off-season and were engaged in discussions to sign Napoli’s Kalidou Koulibaly, while Atletico Madrid claimed the club made a bid for their centre-half Jose Gimenez.
Manchester City were given an extra week off to start their Premier League campaign after their Champions League journey to the quarter-finals.
The Leicester defeat is a significant set-back in their goal to wrestle back the championship from Liverpool, who have won both their league matches against Leeds and Chelsea, and only lost three times in the league last season on their way to the title.