Wilder retained his WBC heavyweight title following a split decision draw with Tyson Fury in Los Angeles at the weekend,
It seems likely that Wilder and Fury will meet for a rematch sometime in 2019 after the controversial decision.
According to his co-manager Shelly Finkel, the Bronze Bomber is very interested in taking on Joshua.
And the Brit took to Twitter to respond to the WBC champion and Fury, asking him why they waited so long to lay down the gauntlet.
He tweeted: “What took this fool so long? Like we ain’t been interested?!!
“Anyway well done Fury! They wanted to get you because they assumed you was finished!!
“I’ll give you a fair one when your ready! Either one of you!”
Joshua is currently the unified world heavyweight champion, holding three of the four major championships.
The 29-year-old knocked out Alexander Povetkin in September to improve his record to 22 wins from 22 fights.
He tweeted a follow up soon after, saying: “Anyway back to the gym – over in USA training local time 18:42! “I’ll get some reply’s done this week #AJBXNG”
Fury has also called Joshua out for what he claims would be the ‘biggest boxing match in British history’.
“We need to give Britain the biggest fight in British boxing history,” the ‘Gypsy King’ said. “That’s me and Joshua.
“The country wants it, I want it, everybody wants it. The only people who don’t seem to want it is their team.
“Let’s get it on, let’s give the fans what they want and have a real battle.”
Fury also took a shot at Joshua on Saturday for not taking the Wilder fight.
“There’s another certain heavyweight out there,” he said before starting to cluck like a chicken.
“Chicken! Chicken! Joshua, where are you? Where are you, AJ?”