Defender Dias becomes the second Portuguese to receive the honour after former Manchester United forward Cristiano Ronaldo.
Centre back Dias was also named England's Footballer of the Year in the annual vote of the Football Writers' Association last month following a superb debut campaign after his move from Benfica last year.
Dias played a key role in delivering City's third Premier League title in four years as well as the League Cup, and helped his side the Champions League final.
The 24-year-old was also named in the Professional Footballers' Association (PFA) Team of the Year.
Dias is the fourth defender to be voted Player of the Season by the public, the captains of the 20 Premier League clubs and a panel of experts after Manchester United's Nemanja Vidic, former City captain Vincent Kompany and Liverpool's Virgil van Dijk.
Other nominees for the award this season included Kevin De Bruyne, Bruno Fernandes, Jack Grealish, Harry Kane, Mason Mount, Mohamed Salah and Tomas Soucek.
Foden made 28 appearances for City, finishing with nine goals and five assists to earn a spot in England's European Championship squad.
The 21-year-old midfielder has also been nominated for the PFA men's Player of the Year alongside Dias, as well as for the Young Player of the Year award, with the winners set to be announced on Sunday.