Wilder retained his WBC belt with a controversial split decision draw against Fury last weekend and the American's team will now discuss options for his next fight, Sky Sports reported.
Wilder's co-manager Shelly Finkel confirmed The Bronze Bomber is "very interested" in a unification fight with Joshua, and the WBA 'super', IBF and WBO champion welcomed the challenge in a fiery statement on Twitter.
Responding to Finkel's comments, Joshua wrote, "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 you're ready! Either one of you!"
Fury came back from two knockdowns, including a heavy fall in the 12th round, to reach the final bell against Wilder.
However, despite appearing the dominate much of the fight, the Gypsy King was denied a famous win after the judges called a draw in Los Angeles.
Meanwhile, Wilder said he "can't wait" to face Fury again.
Wilder claimed he won the fight and was the "more aggressive fighter and landed the more effective punches".
In a post on Instagram, he said, "You saw the best Fury but you did not get the best Wilder and I still managed to get the job done."
He added, "At the end of the day, boxing wins. The fans are the real winner and I can't wait for Wilder Fury 2 to end the controversial talk around the world once and for all."
Saturday's fight was scored 115-111 for Wilder, 114-112 for Fury and 113-113, with Alejandro Rochin the judge who had the American as the winner.
Fury said he had "never seen a worse decision in my life" and described it as a "gift" for his opponent.
His promoter Frank Warren said he and the British Boxing Board of Control would write to the WBC demanding another bout.
Fury was also backed by a number of former world champions, including Floyd Mayweather, Lennox Lewis, Tony Bellew and Carl Froch.