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The IBF-WBA champion can win a third belt when he takes on WBO title-holder Joseph Parker at Cardiff’s Principality Stadium on March 31.
If Joshua wins the Briton would only have to beat American Deontay Wilder to become the first four-belt world heavyweight champion.
The Watford-born boxer, who has KO’d all 20 professional opponents after striking gold at the 2012 Olympics, is already world boxing’s biggest star after American Floyd Mayweather’s retirement.
Sir Henry Cooper, who twice fought Muhammad Ali but never won a world title and died in 2011, is the only British boxer to have been knighted.
But AJ says he hopes his ring exploits can see him follow in the footsteps of Our ‘Enry. He said: “One-hundred per cent I’d fancy it.”
Anthony Joshua and Joseph Parker square off in first press conference
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Joshua, who received an MBE from Prince Charles in 2013, added: “Do you need an MBE to give you the credentials to say you have done well? You already do your part regardless.
“With or without it I will still be an ambassador for the sport.
“If I don’t win a knighthood I’m not going to be ‘for goodness sake’ and start acting a different way. I will be who I am until the end. If I am honoured I’m more than happy to take it.”
Joshua, 28, is more concerned with gaining boxing titles this year than making the Honours’ list, starting with victory over unbeaten Parker.
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The New Zealander plodded to a points win over Manchester’s Hughie Fury in September and Joshua is expected to win and set up another unification clash with WBC champion Wilder, who faces Cuba’s Luis ‘King Kong’ Ortiz on March 3.
Joshua, who lives in north London, wants Wilder at Wembley Stadium this summer after 90,000 saw him KO Wladimir Klitschko at the same venue last year.
With record-breaking support at home, Joshua says he could resist the lure of ever fighting in America. AJ added: “I would love Wilder to be at Wembley – Wembley would be sick for that in the summer.
“I wouldn’t mind going to the States if people were to say, ‘Look, Josh, this will cement your legacy.’ But I don’t want to do it if it was‘Let’s go to Vegas because everyone else has done it’, I don’t want to do it for that.
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“What we did with Klitschko at Wembley was phenomenal. I still train at Finchley ABC so a lot of my support is home-based.
“I always say it gives people an opportunity from my local area to travel half-an-hour to Wembley for one of the biggest fights in boxing history.
“That’s why I don’t want to travel to America because it limits a ton of people from being able to come and watch me.
“Yes I can go through my career not fighting in America because it’s all about where the market is and it’s so strong here. It’s brilliant here, so why would we leave it?”