McIntyre defeated Lesnar at WrestleMania 36 and called out WBC heavyweight champion Fury after his historic victory.
In response, Fury ridiculed Lesnar, whose actual age is 42, and claimed he would have "set about" both wrestlers had he not been prevented from attending the event due to the coronavirus pandemic.
Fury has appeared in WWE before, defeating Braun Strowman at the company's Crown Jewel show in Saudi Arabia last October.
He thinks that experience will stand him in good stead to take on McIntyre, who became the first British world champion in WWE history.
Addressing the challenge on Instagram, Fury wrote: "Drew McIntyre has called me out after his WWE WrestleMania 36 victory.
"First I say congratulations and what a fantastic job. Secondly, I accept his challenge, anytime, any place, anywhere."
Fury then elaborated on the situation in an interview with talkSPORT.
The undefeated boxer said: "I was supposed to be at WrestleMania 36 but obviously with the coronavirus I couldn't be there.
"Drew McIntyre? Well I probably would have jumped in the ring and set about both of them
"I'm all open for it, open for the challenge. I have beaten one big giant wrestler who was bigger than Drew - Braun Strowman, knocked him clean out - I'm sure I'll do the same to our old Drew!
"I'm ecstatic for him because it's the first ever time in history that a British fighter has won the WWE heavyweight championship of the world.
"So congratulations to Drew, but I'm not a 45-year-old man like Brock Lensar, I am in the prime of my life.
"Like I've shown before, I can cross over to any sport and win. If Drew McIntyre wants the Gypsy King then he doesn't have to look very far because I'm here and I'm available!
"[I beat] Deontay Wilder, the most fearsome puncher in the history of heavyweight boxing, so I'll have no objections to smashing Drew McIntyre."
Photo Getty Images Caption: Tyson Fury has previously appeared with WWE