Parker retained his WBO title in Manchester last night with a majority decision win over Hughie Fury, a decision that infuriated the British fighter’s camp.
However, the New Zealander is unlikely to see the result overturned and is now looking to book another big fight in Britain after winning on his first competitive visit.
And his promoter David Higgins says Joshua’s people have already been in touch about the possibility of the two fighters facing off.
“If the Joshua fight is going to sell 100,000 – which it would, the unification – you want it in the English summer, so probably July,” Higgins said in Parker’s dressing room last night.
“I think it makes sense as long as both guys keep winning. Eddie Hearn has reached out and put that on the table.
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“We like to keep busy but this latest mandatory has used up a lot of time. We would have liked to fight a little more often.
“We have a will to fight before the end of the year and there are some good options such as Japan, the United States or going home to New Zealand or Samoa.
“After that, we could look at the winner of Haye vs Bellew, or Joshua.
“If Tyson Fury makes a comeback and Eddie doesn’t give us a fair deal, we could be at Old Trafford against Tyson Fury.”
Former heavyweight champion Fury was ringside last night and called Parker “a p***k” after the decision went against his cousin.
None of Britain’s other heavyweight contenders were in attendance but nevertheless Las Vegas-trained Parker wants to fight more in the UK, while retain his base in the US.
“I want to fight more here because the heavyweight scene is here and it’s booming at the moment so I want to be a part of it,” Parker said.
“I want to get the big fights that are here.
“I want anyone and everyone. Whoever is keen to fight
“But I’ve got to take some time off to relax with the family and chill.
“I said before I’m going to have a pizza and a burger – I think I deserve it!”