Arsenal and Chelsea are also interested in the Ivory Coast international but have not bid close to City's marker.
Toure's agent Dimitri Selguk has met with Barca technical director Txiki Begiristain.
Set to sign: Manchester City are closing in on Ivory Coast midfielder Yaya Toure
He said: 'Within a couple of days I hope we have a solution, but before that I must speak to the player.
'I think Barcelona would prefer this option (City) because they are the ones who are offering the most money.'
No go: Barcelona don't want to sell contract rebel Dani Alves to Manchester City
Barcelona, however, remain reluctant to sell Brazil full-back Daniel Alves to City despite the fact he has rejected their second and latest contract offer.
Meanwhile, City are considering loaning midfielder Vladimir Weiss to Newcastle.
Toure set to complete £25m switch from Barcelona to Manchester CityManchester City hover as Dani Alves rejects Barcelona's latest contract offerMan City edge closer to signing Spain winger Hernandez from ValenciaMANCHESTER CITY FC