"The heavyweight division needs strong and good fighters. Last night I saw a new heavyweight fighter who can fight for the world title."
The Ukrainian said a fight with Haye could happen next spring or summer, in either Great Britain or Germany, where he is based.
However, he thought the Englishman could do with a few more contests under his belt first.
"He has everything he needs. He has power, speed and good reflexes. Two more fights and he will be ready to challenge for a world title," said Klitschko.
One man who could stand in Haye's way of an immediate world title shot against the elder of the two Klitschkos is Lennox Lewis, though a return for the former undisputed heavyweight king looks very remote.
Lewis retired back in 2004 but Klitschko still hopes he can get a re-match with the Briton, who beat him in controversial circumstances in his last bout before retirement.
"I hear Lennox Lewis is back in training and starting to think about his comeback," Klitschko told Sportsweek's Garry Richardson.
"If that fight happens, then Wladimir will fight David Haye. If not, I will be ready to fight David Haye. I am very interested to prove my skills against the new generation."
Haye's trainer Adam Booth seems to be leaning towards the younger Klitschko at the news conference following his man's win over Barrett, highlighting his questionable punch resistance.
"Monte's got a better chin than Wladimir, so if David can nail him, he'll hit the floor," said Booth.
I believe I'm the best fighter in the world, so you can put me in with anyone. I'll get them out of there
"But all of a sudden they're started putting the one with a stronger chin [Vitali] in front of us. That wasn't the plan.
"We'll see what the Klitschko plans are and try to get Wladimir. If Wladimir says 'I'll do it', we'll do it immediately. But we'll take the first one that comes along."
Vitali has only suffered two defeats in 38 fights - on cuts against Lennox Lewis in 2003 and after injuring his shoulder against Chris Byrd in 2000.
Wladimir, meanwhile, has suffered three stoppage defeats in 54 fights and has been knocked down on a number of occasions.
Haye said he would ideally like to fight either Klitschko next spring, but that he would fit one more opponent in between if the fight could not be made until next summer.
"When my title fight comes along I'll be prepared, because I've been in with strong guys who come to win," said Haye, who was floored by Barrett in round five before delivering a stunning knockout blow.
"Vitali feels he's a lot bigger and a lot stronger than me. He's a lump of a man, but I feel I've got his number.
"I've got a strategy to beat him. It's all about speed, I'm not going to have a wrestling match with him.
"I'm definitely faster than him and if I land the shots I landed on Barrett on either of those brothers they're going to know about it.
"And Corrie Sanders was 225lb when he knocked out Wladimir, but he had the speed and skills to get him out of there, so I'm not going to bulk up for the sake of it.
"They're both interested because there are no other exciting challengers for them.
"They want to secure their legacy and who better to fight than the undisputed cruiserweight champion who's stepping up and has a big knockout ratio?
"I believe I'm the best fighter in the world, so you can put me in with anyone. I'll get them out of there."
Haye, who now has 22 wins and 21 knockouts from 23 fights, will be ringside for Wladimir's clash with former world champion Hasim Rahman next month.
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