The Briton successfully defended his International Boxing Federation (IBF) world heavyweight title for the third time and claimed the vacant World Boxing Association (WBA) belt against the former champion after an epic battle was settled by Joshua’s 19th consecutive stoppage.
In a wildly exciting fight, which swung one way and then the other, Joshua sealed victory during a thunderous 11th round.
"I don’t mind fighting him again, if he wants the rematch," Joshua said at a press conference after the fight in front of 90,00 fans at Wembley Stadium.
"Big respect to Wladimir for challenging the young lions of the division.
"It’s up to him, I don’t mind. As long as Rob (McCracken, Joshua's trainer) thinks it’s good I'm good to go."
'One of the best'
Joshua’s promoter Eddie Hearn said that if a rematch were to take place in the UK later this year, it would likely be at the Principality Stadium in Cardiff, which has a roof.
"It was one of the best heavyweight fights I’ve seen and Klitschko will probably say he wants a rematch," Hearn told reporters.
"Whether he feels the same on reflection I don’t know. We’re not opening negotiations with any other fighter until he knows what he wants to do.
"If people don’t want to see a rematch, it won’t happen. If people want to see a rematch, it’s likely to happen.
"It’s more likely to be at (the Principality Stadium] if it’s in the UK, or it might take us to a new territory.
“Anthony wouldn’t mind going to Germany for it, but he won’t be ready before October and that was a bad cut Wlad had.”
Klitschko, 41, refused to make any commitment on his future and whether he would retire after a second successive defeat.
"I’m not going to consider anything, I’m not going to make any statements, I will take my time and let you know," Klitschko said.
"I have a rematch clause in the contract which I can execute. Right now I will not announce anything.
"I don’t feel like someone who lost. Even though I didn’t get the belts, I definitely gained although I lost."
Klitschko suffered his first stoppage defeat since 2004 when he was overwhelmed by Joshua’s precision and power in a ferocious 11th-round assault.
But Klitschko showed there is still plenty of fight left in him.
Joshua had lost control of the epic encounter after flooring the Ukrainian in the fifth and appeared to have punched himself out.
'I can knock out anyone'
Klitschko landed a right to the temple in the sixth round which dropped Joshua for the first knockdown of his professional career.
"I thought he would not get up," said Klitschko. "I thought I could have done more to finish him after he went down. I was pretty sure it was going to be my night so I took my time."
Joshua, the 2012 Olympic gold medallist, became subdued as Klitschko, who won the Olympic title in 1996, clawed his way back into the fight until the Briton unloaded a series of combinations to seal victory.
"I took a round off to get my breath back and then I said: 'You’re in a bad place'," Joshua recalled.
"I tried to recover and then step it up in the later rounds, that was the plan.
"I learned I can knock out anyone. Sometimes you have to ride a few shots and take a few shots before pushing on for victory."
Joshua's powerful right uppercuts featured in all three of the knockdowns but the fight was close at the time of the stoppage: one judge had it 95-93 to Klitschko, while the other two scored it 96-93, 95-93 for Joshua.