Fury had been hoping to make his American debut on St Patrick's Day but instead had to settle for a fight a month later, and his trainer Peter Fury says that they are looking to name Cunningham as the opponent.
"The date of April 20 is now 100%," Fury Snr told WorldBoxingNews.net.
"We are looking at getting Steve Cunningham in or one or two others for Tyson's American debut. A four-month training camp is too long but we are looking forward to getting out there and making a statement in the USA.
"I think a fight with Cunningham is a good fight, especially when everyone says he beat [Tomasz] Adamek. So if Tyson does a good job on him it will send a message to everyone [in the division] and I have no doubts that he will."
Fury defeated Kevin Johnson in his last fight at the beginning of December.