However, he relinquished his titles the following year after his license was suspended pending investigation for doping and medical issues.
Joshua, who became the IBF world heavyweight champion after beating Charles Martin in April 2016, is unbeaten in 18 bouts and will fight Klitschko, who has not fought professionally since his loss to Fury, at Wembley on April 29.
"I think Tyson Fury is the biggest fight for me," Joshua told British media from his training camp in Sheffield.
"If it were up to me, that would have been the fight happening this summer ... I was planning for a massive showdown with Tyson Fury, but he had other plans and I just had to carry on. When he returns, I'm sure I'll still be here."
Joshua and Fury traded barbs on Twitter last week after the former champ called Joshua "a poor man's Frank Bruno," a reference to the former British WBC and European heavyweight champion.
"Twitter is about communication and it's just communicating," Joshua said. "He addresses me and Tweets me quite often and I was just in the frame of mind where I thought me and Klitschko don't have much verbals, so let me respond to Fury."
Joshua added that fury's "Bruno" jibe showed a lack of respect.
"If he's an all-time great then why is he worrying about who is behind him? You should focus on your future," the 27-year-old said.
"If I was the greatest of all time I would never worry about who is behind me. I'm leading the way."
(Reporting by Aditi Prakash in Bengaluru; Editing by Peter Rutherford)
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