The Ukrainian legend hung up his gloves after being stopped by Joshua at Wembley in April 2017.
He has shown no inclination to want to return to boxing but has recently distanced himself from comments suggesting he would never be interested in fighting Joshua for a second time.
And amid uncertainty over Joshua's next opponent, promoter Eddie Hearn says Klitschko has made a personal appeal to the Watford fighter.
He told iFL TV: "He wants to come back.
"I think he text AJ and said: ‘I wanna come back, I feel like I beat everybody in the division right now.’"
Joshua scheduled to fight at Wembley Stadium on April 13, but there is little clarity over the identity of his opponent.
The most likely outcome is the WBA, IBF, and WBO champion will face the winner of Dillian Whyte vs Dereck Chisora on December 22.
Deontay Wilder and Tyson Fury could meet in a rematch of their own in the interim, setting up the tantalising prospect of the victors of both bouts facing off later in 2019.
Reigning WBC title holder Wilder remains Joshua's number one target as he attempts to become the undisputed heavyweight champion.
Talks between both fighters have ebbed and flowed, with negotiations falling through last year leading to Wilder facing Fury in Los Angeles earlier this month.
But Hearn explained: “He has always wanted to fight Deontay Wilder and he wants to fight him now.
"He wants to become the undisputed heavyweight champion of the world.
"The only way either fighter can do that is to fight each other.
"Currently it’s Deontay Wilder, that’s what we want.
"It might be Tyson Fury.
"It might be the winner of Whyte vs Chisora, it might be Oleksandr Usyk, it might be Jarrell Miller.
"Everything’s in play."
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