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.