Price knocked out John McDermott inside a round earlier this month to secure a shot at British champion Fury.
However, that big domestic dust-up may not happen if Fury opts to sidestep Price, ditch the Lonsdale belt and defend his Commonwealth title instead.
Price's promoter Frank Maloney has made an offer to the Fury camp with an outcome expected by the end of the week.
"We have made a formal offer to Fury and they have until Thursday to agree terms," Maloney told Sky Sports.
"We want that fight as quick as possible - he's got to defend it by the end of May but we're looking at April."
Maloney has found it difficult to match the 6ft 8in Price, who has won 10 of his 12 professional outings by stoppage since he joined the paid ranks after winning bronze at the 2008 Olympics.
He has boxed less than seven rounds in the last 12 months and Maloney admitted: "I want to keep him fighting, keep him working."
The promoter added he was 'in talks' with contacts in America should the Fury fight not come off.