Joshua beat Klitschko in 11 rounds on Saturday to unify the IBF, WBA and IBO titles and extend his record to 19 straight wins since turning professional.
And the Brit said on Sunday he would consider a rematch if he received trainer Rob McCracken's blessing.
The defeat to Joshua was Klitschko's fifth defeat in 69 fights and a clause in the pre-fight contract allows the 41-year-old the option to ask for a rematch.
Speaking immediately after the fight, Klitschko said he would take some time before considering a return to the ring.
"A rematch will be good, because of the type of fight we had," Joshua said.
"I think he will want to fight again, because a fighter is the last one to know when to stop. But I think the team around him, his (soon to be) wife, because they normally wear the trousers, and his brother might advise him differently."
Joshua said he respected Klitschko and had no qualms about a rematch.
"I have the ultimate respect for him inside and outside of the ring. I wouldn't mind fighting him again if he wants the rematch. It's up to him... So if he'd like to fight again, no problem," Joshua said.
If Klitschko retires, Joshua could face fellow Briton and former heavyweight champion Tyson Fury, who is yet to regain his boxing license, after the two traded words on Twitter last week.