And the gypsy king is hoping AJ defeats Deontay Wilder should the pair have their unification scrap, so it can set up the all-British showdown.
Fury is making his return to the ring after two-and-a-half years out on June 9 when he fights in his home city of Manchester.
Meanwhile talks continue between Joshua and Wilder's teams in the hope a deal can be reached for an 'all the belts' heavyweight blockbuster.
And while Fury isn't bothered who of the two he fights as he would "beat them all", he is keen to take AJ's unbeaten record.
Speaking to SunSport, Fury said: "Deep down I hope it is Anthony Joshua, because he is a Brit, and he is a world champion.
"He has done very well, and I would like to be the one to take that unbeaten record off him.
"It would be a massive sporting event in this country, and an event all boxing fans would be looking forward to so much.
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"It would be a crying shame if he [Joshua] were to go and lose to Wilder and have the Americans stealing our thunder again.
"I’d have to go over there and take it back. It’s not ideal, it’s not what I would want to have to do."
And the Gypsy King wants to make sure he puts on a real show for the British boxing fans in a potential fight where someone's 0 has to go.
He added: "I’d rather have the fight here in the capital, not so far away. I want to put some great shows on in Britain.
"I can guarantee it will be an exciting fight, exciting press conferences, exciting build up.
"It will be an event that will take us back to the World Cup in 1966, an event people will be talking about in 50 years to come.
"The fight will have to happen because this town ain’t big enough for the both of us."
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