Fury had been hoping for a St Patrick's Day fight on 17 March, but the bout has been pushed back to 20 April and Cunningham could be in the opposite corner.
The American dropped a disputed points defeat to Tomasz Adamek in December and the Fury camp want to make a big impact in the US with a convincing win over Cunningham.
Trainer Peter Fury told World Boxing News: "The date of April 20 is now 100% for Tyson and we are looking at getting Steve Cunningham or one of two others for Tyson's American debut.
"We have been in training now for a March fight so we tried to get a fight in between in Germany, but (promoter Mick) Hennessy said not possible so we will be easing Tyson back a bit.
"A four month 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 Adamek. So if Tyson does a good job on him, it will send a message to everyone and have no doubts he will do."