Reports last week suggested Toure could be part of a summer clear-out at the Etihad Stadium as the club look to reshape the squad after an underwhelming season.
That was denied by manager Manuel Pellegrini, who insisted Toure would “continue playing here because he is a very important player”.
But Pellegrini did acknowledge Toure’s form this season has been disappointing and speculation linking the Ivory Coast international with a move away has not subsided.
Now the 31-year-old has given an interview suggesting he could be open to a move.
Toure, who is reportedly City’s highest earner on £220,000 (€305,000) per week, told Foot Mercato: “No amount of wages will make me stay at a club if I feel I no longer belong there or if no challenge exists for me.
“It would be unjust on my part. There comes a moment where numbers don’t stop us, it goes beyond that, even if the English press seem more interested in the numbers than the sport itself.
“For the future, I don’t know more than you do, because I will always go where I am offered new challenges. That is in my nature.”
Despite those remarks, Toure insisted he remained fully committed to City on the field.
The two-time Premier League winner added: “When I arrived at City, Pellegrini was not the coach. Just like the players, managers arrive and leave. As I have said before, I owe it to the City fans to fight until the end of my career at this club.
“My decisions will not be affected by changes in management, but more by the challenges that will be offered to me.”
Toure has been linked with Inter Milan, whose manager, the former City boss Roberto Mancini, has spoken openly and regularly of his admiration for the player.
Toure says the feeling is mutual.
He said: “Mancini is a mentor for me, and a coach out of the ordinary. It is no secret that I loved when he was my boss, as I like to play for City today.”
Toure is one of a number of players to have faced criticism for City’s disappointing campaign.
City have collapsed in the Premier League title race –— slumping from joint-top in Janauary to fourth –— and failed in the knockout competitions.
Toure said: “When things are not necessarily going well in a club, the key players take the fall.
“I am not the only one to have been attacked even if there is tendency to be harsher with me.
“Football is my passion, my job and that gives me two good reasons to do as well as I can.”