The goalkeeper and striker will follow Vincent Kompany, David Silva and Sergio Aguero in agreeing to fresh terms but there are no talks scheduled with Manuel Pellegrini.
The Premier League champion has already secured the long-term futures of David Silva, Vincent Kompany and Sergio Aguero in the last week, as well as signing Eliaquiam Mangala and Fernando from Porto, Bacary Sagna from Arsenal, and Willy Caballero from Malaga during the transfer window.
England goalkeeper Hart will face stiff competition to secure his status as City's No. 1 this season following the arrival of Caballero but is ready to commit his future to the club regardless.
Caballero started against Arsenal in the 3-0 Community Shield defeat last weekend and could do so Sunday as City begins the defense of its title at Newcastle after Hart celebrated the birth of his first child Friday.
Dzeko, meanwhile, looked set to leave the club this time last season following the additions of Stevan Jovetic and Alvaro Negredo but has flourished under Pellegrini with goals in three of last season's final four matches helping to secure the title.
A source told Goal: "The club's owners are of the opinion that a manager's cycle should last three years. They do not want a repeat of the scenario whereby Roberto Mancini was offered a five-year deal after winning the title, only for him to receive a huge payoff just 12 months later."
Bruno Zuculini, meanwhile, is close to agreeing to a season-long loan move to Valencia. The two clubs held talks Saturday and the young Argentine is expected to complete the switch early next week.