But he has avoided the Kiwi star for the time being by opting to go after Martin.
Parker is now set to announce a fight in Christchurch in late April.
Joshua, who has knocked out all 15 of his opponents, said: "Fighting for the heavyweight world title has been a dream of mine since I turned professional.
"I feel privileged to have the opportunity to turn that dream in to reality. Not only that, having the chance to fight for it in my home town is unbelievably exciting.
"Charles Martin is a great fighter and a hungry competitor so I am going to have to produce the performance of my career to claim that belt."
Meanwhile, Martin said: 'I told my Team I wanted the best challenger possible for my first defence and Anthony Joshua's name kept coming up.
"I'm coming to the UK to make a statement that I am the best heavyweight in the world and no-one is taking my title.
"I'm World Champion, so that doesn't mean just sit back and make easy defences in the US, it means facing the biggest challenges out there. I respect Anthony Joshua and everything he has accomplished, but on April 9 he's getting knocked out."