Real have caved into Jose Mourinho's demands for a new attacking signing and the coach wants a deal for Adebayor.
Jorge Valdano, Real's GM, said: "We are exploring the market now the uncertainty around Gonzalo Higuain has been resolved. We'll see if we can find a player who can help us for six months.
"Many names are being mentioned and it's best not to go into details. But it's true we are looking for a No 9."
City are understood to be willing to allow Adebayor to leave, as long as the majority of his £160,000-a-week wages are covered.