The WBA world welterweight champion is taking on Robert Guerrero in Las Vegas on Saturday night and is likely to be the mandatory challenger to the winner of Floyd Mayweather or Manny Pacquiao.
Thurman believes Khan has not done enough in the 140lbs division to be at the front of the queue after May 2, and is more than happy to take on Brook.
“That’s become a more frequently asked question, especially ‘Will I go over there?’,” he said.
“Well, I would love to. I love travelling the world of boxing. At this time, I don’t know what plans they (Brook’s promoters) have and I don’t know what they (Thurman’s promoters) have in store for me, after this fight on Saturday night.
“I am truly looking forward to Saturday night and coming through it but as long as Kell is a champion and I am a champion, it is definitely a fight that I am willing to make.
“That could possibly be made.”
Brook first has to make the first defence of his IBF title against Jo Jo Dan on March 28, live on Sky Sports.
His promoter Eddie Hearn says discussions are already underway for him to take on Khan in an all-British showdown.
Thurman admitted he would like to see that as well because the two-weight world champion from Bolton needs to prove he is as good as people say.
“Amir Khan is still overrated because he is yet to face a a true welterweight,” he said.
“He keeps picking the right fights and the right time. Devon Alexander? They could have both fought at 140lbs a year-and-a-half ago. But fighting at 147? That is not a welterweight fight.
“He needs to fight Shawn Porter or Kell Brook. If he wants to make a statement in the welterweight division, he should fight myself. There are several ways he can make a statement.”
The clash between Thurman (24-0 KO21) and former super-featherweight world champion Guerrero (32-2-KO18) is topping the bill at the MGM Grand on Saturday night, the first show of Al Haymon’s World Professional Championship.
Adrien Broner against John Molina Jr and Abner Mares against Arturo Santos Reyes are the major supports.