Former champion Fury fought to a thrilling, but controversial, split decision draw with American WBC belt holder Deontay Wilder in Los Angeles last month and a re-match looks likely.
"He's the people's champion," Warren said of the self-styled 'Gypsy King'. "I think he's now a bigger attraction than Anthony Joshua."
"Whoever he fights next, if Tyson Fury fights in this country, it is going to be a sell-out, wherever he fights," he told the BBC.
"At the moment I'm talking with Shelly Finkel, who looks after Wilder, and we are trying to get the rematch together."
Fury's bout with Wilder was only his third since a two-and-a-half year absence from the sport following mental health issues and a failed drug test.
Joshua is the undefeated holder of the WBA, IBF, WBO and IBO heavyweight belts and a unification bout with Wilder would be a blockbuster draw.
His next fight is due at Wembley Stadium on April 13 against an as-yet unnamed opponent.
"Tyson only wants big fights. There's only two big fights out there -- that's Wilder, and that's Joshua," said Warren. "Joshua doesn't even want to fight him because I think he knows he'll get beat."
Warren said a Fury v Joshua fight, possibly in Cardiff, would sell out 10 times over.
"It would grip the country -- one of those events that transcends the back pages. Everybody would be talking about it," said the promoter.
(Reporting by Alan Baldwin, editing by Toby Davis)
