Heavyweight Joshua was initially due to defend his IBF, WBA and WBO titles against Jarrell Miller in his American debut, with the bout set for Madison Square Garden in New York.
However, Miller was ruled out after a second failed drugs test, leaving promoter Eddie Hearn needing to find a replacement for the main event.
Speaking at a press conference to promote Dillian Whyte's July fight with Oscar Rivas, Barry Hearn, Eddie's father, declared they are "very close" to confirming the identity of Joshua's next foe.
"I know he [Eddie Hearn] is very close," Barry Hearn told Sky Sports.
"I understand there's every chance he's going to make an announcement tomorrow [Tuesday], so it's getting there.
"It's difficult, it's very difficult, the political world of boxing and the money they're talking about, but it's also difficult when a few people view it as an opportunity to put one over on AJ, or us, or boxing."
Always representing. Finchley ABC pic.twitter.com/D3pKblTiib
— Anthony Joshua (@anthonyfjoshua) April 29, 2019
Barry Hearn also revealed that they had created a shortlist of candidates, with Joshua having the final say.
"We have five likely opponents, so it's not a question we're going to be blackmailed. We're never going to be forced into doing anything stupid, because we just don't work like that. Eddie is just playing the cards dead straight," he added.
"AJ makes the final call, the final list is there and I believe the final decision will be known tomorrow."
Luis Ortiz ruled himself out of contention via Instagram last week, revealing "low-ball offers" from Matchroom Boxing had ended any chance of him stepping in to fill the void.
Photo: Richard Heathcote/Getty Images. Caption: Anthony Joshua has the final say on who he will face on June 1.