Brook has agreed to step up two weight divisions in order to face the unified world middleweight champion at London's O2 Arena, and the move has left Khan open to facing the Sheffield fighter at some point in the future.
Khan had previously stated that he would only agree to a showdown with Brook should he face better opposition, and the 2004 Olympic silver medallist has indicated that he is open to starting negotiations sooner rather than later.
The 29-year-old told the Dubai Press: "I have to give him props for taking the fight [against Golovkin]. It's a very hard fight so I give him a lot of credit. He can win... well actually, no, I don't think he can.
"Certainly I still think a fight between myself and Kell will happen because he has proven himself now by being willing to step into the ring with Golovkin.
"I still don't like him, in fact I hate him. But the UK public want to see that fight and it is a fight I want to give them."
Earlier this year, Khan made a similar step up in weight to face Saul Alvarez but he was knocked out by the Mexican in the sixth round.