The pair are expected to go head to head in February,
Ben Shalom, the chief executive of promoters Boxxer, says an official announcement of the all-Brits clash is expected on Monday.
Shalom told talkSPORT: “It’s no secret, for the past two months we’ve been trying to make Amir Khan against Kell Brook.
“I think it’s a fight that every boxing fan has wanted to see for years and years and there’s been a lot of competition – every promoter has wanted it, no matter what’s being said and what isn’t being said.
“I think we are very, very close now and hopefully we’ll announce that on Monday and it will take place in February in Manchester.”
Former WBA and IBF light-welterweight champion Khan, 34, last fought in July 2019 when a victory over Billy Dib took his record to 34-5 (21 KO's).
Brook, an ex-IBF welterweight title holder, was most recently in action with defeat to WBO champion Terence Crawford in November 2020. The 35-year-old’s record is 39-3 (27 KO's).
Shalom added: “It’s the biggest fight Boxxer have been involved in.
“Kell Brook fought for a world title last year and looked good up until being stopped by perhaps the pound-for-pound king.
“These guys are ages 34, 35 – you almost get that direct comparison to the (Manny) Pacquiao-(Floyd) Mayweather fight.
“These guys would still compete extremely well domestically. I think they’re both at a similar age, similar point in their career, and if the fight does happen, which it looks very, very close to be happening now, it’s something that I just can’t believe we’re involved in.”
Earlier on Saturday, a post appeared on Khan’s official Twitter feed, featuring footage of him training in the gym, and the message: “Fight news coming soon. Khan v ?”