Both fighters’ promoter Eddie Hearn believes a fight between the two British boxers will happen next year, most likely in February or March.
Khan returned to the ring in April 2018 but easily overcame Phil Lo Greco, knocking out his veteran opponent in 39 seconds.
After the bout, Brook then came into the ring to trash talk the Bolton fighter, but Khan seemed largely uninterested in entertaining a match-up.
Brook was then ringside for Khan’s latest win over Samuel Vargas, when he knocked the Colombian over twice despite being dropped once himself.
British boxing fans have demanded a fight between the two for years, but Khan outlined his goal to fight Manny Pacquiao next, although those plans are looking less likely to come to fruition.
If Khan vs Brook is to happen, the latter has accepted he will have to drop down to 147lbs and have the fight at welterweight.
Although the bout would not be a title contest, Khan has claimed that Brook agreed to a 10lbs rehydration clause in any potential contract.
That means that both fighters must weigh in below 147lbs on the day before the fight, but cannot weigh over 157lbs on a fight-day weigh in.
That is common practice with the IBF, the organisation with which Brook was formerly a welterweight world champion, but this domestic bout means the Sheffield scrapper is under no obligation to follow that rule.
Speaking to Behind The Gloves, Brook said: “Everybody in the world knows what it does to me to make 147lbs.
“He says that he sparred with me many years ago and he used one hand to destroy me. I’m a very, very easy opponent for him.
“He wants me to do 147lbs, he wants me to only put 10lbs on, even though there’s no world title on the line, there’s no IBF belt on the line.”
Brook then opened fire on Khan, who is seen by many as trying to dodge the fight.
“Listen, bottom line, he’s a f**** hole, he’s a little b****,” Brook asserted.
“He’s gonna have to take the fight because everyone’s looking at him now and thinking: ‘What are you talking about?’
“He’s making things up, saying I agreed on the 10lbs thing. Why would I agree on that? It’s a joke. He is a joke.
“I’m the welterweight king in Britain, so don’t claim that you are unless you get in there with me.”