The Ivorian, who moved to the Premier League club from Barcelona in 2010, turned 31 on May 13.
Dimitri Seluk, who has previously criticised City for a lack of respect for Toure, has claimed that the player could leave the club following the reported snub.
"There is a big possibility Yaya will leave this summer. He is very upset. What happened at his birthday means the club don't care about him. It was proof," Seluk told The Sun.
"They can say whatever they want, 'Oh we like him but we forgot'. Forgot? About Yaya? The player who helped you to be two-times Premier League champions? The guy who helped you win the FA Cup and League Cup?
"In a normal team, his team-mates and club would at least give him good wishes. But when you win the title and two days later you have a big party to celebrate the title and nobody comes to Yaya to say happy birthday, what more do you want?"
Toure has three years remaining on his current deal with City.