The 29-year-old England international, who had 12 months remaining on his contract, will end a four-year spell at the Etihad Stadium.
Delph was restricted to just 11 Premier League appearances last season for City, who paid a club-record £62.8m fee for Atletico Madrid midfielder Rodri earlier this month.
Former Leeds and Aston Villa player Delph started for England in their Nations League Finals games against Holland and Switzerland in June.
He won two Premier League titles with City and won his 20th senior England cap in the penalty shootout win against Switzerland.
Everton have signed goalkeeper Jonas Lossl from Huddersfield and Portugal midfielder Andre Gomes from Barcelona in a £22m deal since the end of the season.