The 22-year-old has been linked with a move away from Barça for some time with Paris Saint-Germain having been heavily linked.
Dembele was reportedly part of a proposal that Barcelona suggested to PSG when they tried to re-sign Neymar last summer.
Former Barcelona coach Pep Guardiola is reportedly a huge fan of Dembele's and thinks that, in his City side, the winger can reach the heights his potential promised.
Guardiola is understood to see Dembele in the same light as Raheem Sterling of a few years ago - and will make it his mission to make sure he excels.
It's thought that Barcelona will ask that City pay at least £86m if they want Dembele.
Whilst money is no object for City, they will expect to recoup some funds from the possible sale of Leeroy Sane.
Sane is out of action with a long-term injury issue, but he was heavily linked to Bayern Munich during the summer transfer window.
The German is said to have grown frustrated with his lack of consistent first-team football under Guardiola.
In a similar vane, Dembele has struggled to hold down a starting place under Barcelona boss Ernesto Valverde.
The signing of Antoine Griezmann has made it even harder for Dembele to make an impression.
Since his arrival at the Nou Camp, Dembele has scored 12 goals in 42 games.
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