Retired judge Daniel Weinstein upheld a claim lodged by Wilder that Fury must face him in a third fight by September 15 following their February 2020 title battle.
While the arbitrator’s ruling is not a formal court order, it would be deemed enforceable by courts if Wilder and Fury are unable to reach agreement.
Fury’s promoter Frank Warren told the BBC on Tuesday he hoped Wilder would, for a lucrative fee, agree to step aside and allow the bout to go ahead.
“Until we can reach an accommodation with Deontay Wilder’s people, I can’t say that (Fury’s fight with Joshua) will definitely happen,” said Warren.
“But we are working hard to get that over the line.
“We were talking last night and hopefully we can do that.
“If not then Tyson, if he wishes, will go to the fight with Wilder. They’re the only two options.”
Anthony Joshua’s promoter Eddie Hearn was left fuming by the latest glitch, blaming Fury’s camp for being complacent over the arbitration hearing.
“We can’t wait around,” Hearn said on his company Matchroom’s social media channels.
“We had a deal in place with Tyson Fury and we were told the arbitration wouldn’t be an issue, that we could move on with this fight.
“They were wrong and that’s on them, that’s their responsibility and their problem.
“We hope they can solve that problem, but we have to look after ourselves and Anthony Joshua.
“We have to maintain the position of unified world champion, and those talks will continue and we want to be in a position by the end of this week to know, are we fighting Tyson Fury or are we moving forward with another option.”
Weinstein’s ruling on Monday did not grant Wilder any monetary damages and also said an extension to the September 15 deadline was permissible if both the Wilder and Fury camps could reach agreement.
Wilder’s camp had argued their fighter had a contractual right to a third fight with Fury.
But after delays caused by the coronavirus pandemic led to last July’s contest being postponed, Wilder’s team sought to move the bout to 2021.
Fury refused the 2021 date and said he would move on which prompted the mediation.
Neither Fury nor Wilder have fought since clashing in Las Vegas in February last year, where Fury outclassed the previously unbeaten American to win via a seventh-round technical knockout.
The two fighters had previously clashed at the Staples Center in Los Angeles in 2018, battling to an epic draw after Fury survived a 12th round knockdown.