Saudi Arabia hosted Joshua's previous outing, the heavyweight regaining the IBF, WBA and WBO titles with a unanimous points win in his rematch with Andy Ruiz Jr in December.
Having avenged the only defeat of his professional career at the end of last year, the 30-year-old is now keen for a homecoming in 2020.
His last bout on English soil was back in September 2018, when he stopped Alexander Povetkin at Wembley Stadium. The initial encounter with Ruiz took place in New York the following June, at the famous Madison Square Garden.
Joshua's clash with IBF mandatory challenger Pulev in London is now close to being agreed, according to promoter Hearn, with the pair set to meet at Spurs' impressive new home.
"We're very close. I had a meeting with AJ last night," Hearn told Sky Sports News. "We've had offers in from the Far East, Middle East, Africa, America, Turkey.
"But he [Joshua] has made it very clear to me: 'I want to come home. I want to box in London next. I've been to Madison Square Garden, I've been to Saudi Arabia, bring me home. Forget the other offers, bring me home.'
"He wants to fight in London in June. We are on the verge of making that happen now.
"Spurs is the front runner and that's what he's asked me to do. We'll be delivering that for him."
Photo: Richard Heathcote/Getty Images Caption: Anthony Joshua is set to fight on home soil next.