Brook was ringside as Khan survived a second-round knockdown to claim a unanimous points decision over Samuel Vargas at Arena Birmingham.
The Bolton fighter floored his Colombian opponent early on in the second round, but was then caught with a flush right to the chin, from which he hit the canvas.
He steadied the ship and registered another knockdown in the third, however, eventually coming through 119-108 119-109 118-110 on the judges’ scorecards.
In his post-fight interview, Khan named WBA welterweight champion Manny Pacquiao as his next target.
There is huge demand in the UK for Khan to face Brook, however – a fight which the latter wants.
“I’ve come here to stir things up,” Brook told Kugan Cassius of iFL TV.
“We’re both with the same team [Matchroom Sport]. Let’s have a Christmas special – December, me and Khan.”
Khan signed with promotion company Matchroom Sport in January 2018 following a two-year lay-off from the sport.
The 31-year-old chased Pacquiao and Floyd Mayweather Jr fights during his mid-20s, but never secured them.
Asked whether Brook had eyes on the Filipino, Brook replied: “I’m only after [the Khan] fight… Or Pacquiao.
“I’ve been with Matchroom the longest [seven years] – there’s no way he’s getting that fight over me.
“I’ll go 147lbs [welterweight] and fight Pacquiao. Why should Amir get the fight? I’ve been with Matchroom forever.
“If he doesn’t want to take the fight [against me] because he’s scared, or whatever, I’ll fight Pacquiao. Not him – me.
“People need to stop asking me the question: ‘When’s [Khan v Brook] going to happen?’ Ask him. I’d love to get inside [the ring and ask].
“I’ll put him on his backside if we meet.”