Brook retained his IBF welterweight title with a sixth-round stoppage of Frankie Gavin in London on Saturday night.
Yet, while Brummie Gavin jumped at the chance to challenge the Sheffield champion, fellow British welterweight Khan has so far avoided a huge domestic showdown.
Promoter Eddie Hearn insists he has done all he can to make the all-British fight happen but claims Khan – or his team – are too afraid of losing the America-based cash cow if he lost to Brook.
“I’d love the Amir Khan fight but you lose the passion to talk about the fight,” Hearn said.
“I can’t even be bothered to hype the fight anymore because it seems too far away from being made.
“Khan is a brilliant fighter. You can’t knock him. But you get to this stage and you think ‘you’re just too scared to fight Kell Brook’.
“I’m sorry, but it’s the truth. I don’t mean physically scared, I mean scared of losing.
“We’re going to miss out on this fight happening and one day we’re going to sit down when we’re old and say ‘it was a shame that fight never happened’.
“Khan says it’s not about the money. Then he says it’s not about fighting for Kell’s world title. Well what is it about? He then says ‘it’s all about the legacy’.
“Well, in that case, come and have a major fight at an outdoor stadium in Britain. That’s a legacy, isn’t it?”
Hearn added: “But there are loads of other fights out there for Kell. The winner of Tim Bradley v Jesse Vargas, or Keith Thurman. But we definitely need a big name for Kell here in August.
“29 August is the date we’re looking at. I’d love to do it at Bramall Lane and we’re in talks with Sheffield United.
“Rios is still my first choice because I think he’s the perfect opponent in terms of style, name and entertainment. They seem to want the fight.”
Brook barely spoke of Khan on Saturday, so sick is the Yorkshireman of the speculation and rivalry without the fight being made. Instead, he is focused on the likes of Timothy Bradley, Keith Thurman, Manny Pacquiao and, eventually perhaps, Floyd Mayweather.
“I’m bored of the Khan talk,” he admitted. “I’ve got a world title and it’s a big fight but he doesn’t seem interested. At least Frankie had it in him to get in the ring with me.
“I’m ready for big names now. Rios is perfect for me and a chance to put on a show.”
Brook took time to wear down the slick, elusive Gavin before putting the pressure on in the sixth and forcing the stoppage.
On the undercard, Welshman Lee Selby claimed the IBF featherweight title with a classy eighth-round technical decision win over Evgeny Gradovich but Dagenham fighter Kevin Mitchell was unlucky to be bludgeoned to a bloody tenth-round stoppage defeat by WBC lightweight champion Jorge Linares.
Olympic gold medallist Anthony Joshua stepped up his heavyweight apprenticeship with a fearsome second-round stoppage win over former world title challenger Kevin Johnson.