Toure's agent has previously hinted that the midfielder would be open to leaving the Citizens in the summer.
The Sun reports that the Ivory Coast midfielder's agent has claimed that they will listen to offers for the 32-year-old at the end of the season, and thus, City appear keen to ensure that they have a replacement swiftly lined up.
Barkley's performance levels have arguably gone up a notch this season, as while his talent has always been unquestionable, he is starting to back it up with goals and assists for the Toffees. He has scored six goals and bagged seven assists in the Premier League this term.
City were told when they tried to sign him in 2014 that they would need to part with £50m to take him to Eastlands (via The Sun), and with the form he has showcased during this campaign, it seems likely that the Merseyside club will want even more for the creative midfielder now.
Barkley's biggest flaw currently is arguably his consistency. On a few occasions this season, he has struggled to make anywhere near the kind of impact he is capable of making, and that may concern the Citizens, especially given the amount of money they may have to part with to complete a deal.