New Zealand heavyweight Joseph Parker will wait to see if he could be the man Fury wants after expressing a willingness to take on the British giant.
The WBC had given both camps until Wednesday to confirm their desire for an instant rematch and while Wilder wanted the fight, Fury didn't - for the moment.
They fought a thrilling draw in Los Angeles last December where Fury survived two knockdowns to see out the final bell in a bout many critics felt he had won with some outstanding skills.
Fury's stance backs up the touted theories that his new mega-deal with American promoter Bob Arum and allegiance to broadcaster ESPN could see the title fight sequel put on hold.
Arum feels it might be better to let the rematch bubble away with Fury taking another fight first.
Parker has been mentioned as a possible opponent for Fury along with with Bulgarian Kubrat Pulev, Colombian Oscar Rivas, British contender Dillian Whyte and American Jarrell Miller.
The Furys have history with both Parker and Pulev who have beaten Tyson's cousin Hughie.
Pulev is scheduled for a March 23 fight against Romanian Bogdan Dinu in California but Parker is looking for an opponent to kick of his 2019 campaign.
Parker's promoter David Higgins confirmed they would be keen on the fight if the money was right.
Higgins acknowledged the massive threat of Fury but felt it would generate "huge revenue" and they would accept it if there was "a substantial offer".
Higgins said on Wednesday there had been no contact from the Fury camp.