The location of the highly-anticipated unification bout remains one of the key points to completing negotiations for the heavyweight division's super fight.
Joshua confirmed the fight deal was 95 per cent over the line for a clash early this year.
Parker's promoter David Higgins is having the contract examined by legal experts and is likely to fly to London late this week to sign it off.
Parker could join him in the UK for a press conference the following week.
While Britain remains the favoured country, Joshua's promoter Eddie Hearn revealed he and Higgins were still discussing some "quirky" alternatives.
London's Twickenham Stadium and Manchester's Old Trafford have been mentioned as options, but everything points to Cardiff's Principality Stadium which holds 78,000 people and has a retractable roof to offset the unpredictable British weather.
Joshua is determined to continue to fight in Britain where he has a massive fan base that transcends boxing.
"It has to be (in this country), the British support, what we're achieving here is phenomenal," Joshua told the BBC as hype builds around the fight where he puts his IBF and WBA belts up for grabs along with Parker's WBO title.
"And I always remember when I was watching boxing, it was always about America, America, America and even being a world champion you have to fight around the world, but it's nice to own the market here. We own the market and we're not rushing anywhere, let the British fans stay on home soil for a bit, let other people travel."
Joshua said talks were making pleasing progress.
"I think we're 95 percent of the way there with completing with Joseph Parker," Joshua said, adding that he felt the date would be at the "end of March".