Joshua vowed to knock WBC champion Wilder "spark out" after beating Joseph Parker in Cardiff on Saturday to add the WBO belt to his IBF and WBA straps.
Finkel said the Wilder camp are prepared to come to an agreement to face the Briton, with reports that a two-fight deal for bouts in the UK and the United States is on the cards.
"We are really glad that Anthony Joshua said for the first time, 'I will fight Deontay next...' and we believe he is a man of his word. Deontay accepts the challenge," Finkel told The Daily Telegraph.
"We want to make it public that Deontay wants the fight, and we are prepared to come over, or meet here and get the deal done. Deontay is ready to sign and come to the UK to fight this summer.
"We're ready. Let's do it. And anyway, unification fights always trump mandatories. A unification between Wilder and Joshua would push the mandatories back.
"Assuming it will be Wembley Stadium in the summer, we are ready, and if they are ready as they say they are, we want to get it done. We want the fight, they want the fight, the public wants the fight. There is nothing to stop it going ahead."
Matchroom Sport chairman Barry Hearn said he expects talks to start this week and Finkel sees no reason why the two parties cannot agree terms in the near future.
He added: "We can get the broad contracts down, and then go over the finer details. When I had [Mike] Tyson and [Evander] Holyfield, I picked up the phone. That's all we need to do here. If we get two great fights, there will no doubt be a third. That's the bottom line.
"If this was any other sport, the two leading players would be competing to find out the world number one, and we have a huge opportunity to do this in boxing's great, historic division, and there's nothing in the way off it happening.
"These are two undefeated heavyweight champions. It has been a long time since we've had this opportunity.
"We are prepared to meet as soon as possible, anywhere, anytime, and when I meet Matchroom I will have the authority to make the deal. It's for the better for everybody, and the sport. Unless someone has another agenda. We are ready. Let's make the fight."