The Sheffield ace, known as Special K, issued the challenge to Khan after he dismissed the boxer's claims – made in a recorded interview – he was gay as "rubbish".
In the interview, Khan also suggested his foe was stabbed in the leg six years ago because he "tried it on with some poor guy" and branded his most recent opponent "a stripper".
The Olympic silver medallist was speaking at a charity event in London last Monday just days before Brook took on Aussie fighter Michael Zerafa.
He said: "How much confidence does this guy have? I’ve been told by numerous guys that he’s not a confident guy… There’s talks about him being gay.
"If you’re a real man, come out and say ‘Yeah, you know, I’m gay, and it is what it is.’ Come out and tell the truth. Speak out about it. Don’t be afraid."
Speaking to a journalist from sports agency Hayters, he added: "If he is a confident man, he needs to come out and tell the world who he really is. Because people want to know about him. People want to know, ‘Who is this Kell Brook?’ Come out and tell us what happened.
“What happened when you got stabbed that time? Was it because you tried it on with some poor guy? You tried to get it on with some poor guy? These are things I’m being told from his own camp.
“You know, I wasn’t there to see it. But I’ve been told. He’s living a crazy life. So maybe that’s why he’s dying for this fight against me. Because he knows he needs that quick payday.”
Sources revealed to the Sun Online Kell, whose wife is currently expecting their third child, has slammed the claims as "obviously rubbish".
The source said: "Kell Brook is not making any comment on this because he would rather concentrate on getting the fight arranged and knocking him out of the ring."
Insiders told The Sun Online the video, which has apparently spread online on boxing forums, had left people puzzled and questioning "what on earth is Amir Khan talking about?".
They also pointed to what they said were undeniable facts about the stabbing, which they believe proved the claims were bogus.
Kell Brook's leg is still scarred after a knife attack when he was the world boxing champ while he was on holiday with his pregnant wife in Tenerife in 2014.
He was stabbed in the leg and Spanish cops released a statement saying he’d “lost a lot of blood”.
The boxer needed 32 metal staples to hold his left thigh together and later told reporters he was the victim of an unprovoked machete attack in the early hours by a man he’d met earlier that night.
It was the second time the boxer had been stabbed, having also been knifed in hometown Sheffield in 2007.
In the latest interview at citizenm Tower of London hotel, Khan also tore into Kell's opponent Zerafa before the fight, saying: "I wish him (Brook) all the best. I know he’s going to win the fight.
"There’s no way he’s going to get beat by a stripper. No disrespect against a stripper."
Aussie Zerafa has previously defended himself against the claims, saying a photo of him dressed as a Chippendale was from a fancy dress party
The build up for a fight between Khan and Brook has lasted for three years, with the match having appeared ready to be signed off for early next year.
However, the match was again postponed after Khan revealed he had been offered to fight American Terence Crawford – with reports he was offered £3.9million for the stoush.
The 32-year-old has two daughters with wife Lindsey.
Both Brook and Khan have won two tune-up fights each this year, with Sheffield fighter Brook bagging a points win on Saturday night.
Brook, 32, battered super- welterweight Zerafa into a bloody mess at the Sheffield Arena, with his nose appearing broken as early as the second round.
But Brook was also hit far too often in the second half of the fight — and looked miles off the elite level the former IBF welterweight champ insists he can still mix it at.
Khan watched the WBA eliminator on TV and was far from impressed.
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