The Ivory Coast midfielder was expected to seal a move earlier this month, but the player's participation in the World Cup and the change of presidents at Barcelona have slowed down the transfer.
New Barca chief Sandro Rosell takes over from outgoing president Joan Laporta on Thursday and the move should be finalised shortly after that.
"I think we will have a solution this week. In fact, I would say that (it will be done) on the first or second of July - maybe on the second because the first will be a very busy day at Barca with the change of president," agent Dimitri Seluk told Sport.
Seluk revealed he will speak to Rosell, but insisted his player's mind is made up to leave Barca and join City.
"Yes, we have to talk, but of course, the reality is not going to change in one moment," he said.
"Yaya wants to play the important games and he knows that he will be an important player at Manchester City.
"It is not a money problem, as I have said 1,000 times, but a problem of (the lack of) confidence (that Barca have in him)."
Seluk said nothing had been signed yet, but claimed Toure's salary demands were reasonable.
"I haven't signed anything. However, Yaya doesn't have an astronomical salary, not at all, so it won't be difficult to be reach an agreement," he said.