Wilder retained his belt on Saturday after a thrilling 12-round battle ended in a split decision draw at the Staples Center.
The result means the American is still in a position to achieve his dream of unifying the heavyweight division against Fury's fellow Briton Anthony Joshua, who holds the WBA, IBF and WBO versions of the title.
After the bout, Wilder said he would prefer to target Joshua's titles in a bid to unify the division but would be satisfied with either a rematch against Fury or a chance to take on Joshua.
But Fury said that Wilder could try to avoid him.
"I've got a funny feeling he's going to get his running shoes out and stay away from me at all costs," he told Britain's ITV network.
"I'm only getting better, and better and better. I can beat them all but they need to fight me. We need to give Britain the biggest fight in British boxing history and that's me and Joshua.
"Let's make it happen. The country wants it, I want it, everybody wants it. The only people who don't seem to want it is their team," he added.
Wilder's trainer Jay Deas told BBC radio he would want Wilder to take on Fury again before a potential fight with Joshua.
"If it's something that people would like to see again we would absolutely love to do it. Both guys put on a tremendous fight," Deas said.
"If it was up to me -- and only me -- I would like to do the Fury fight first because I think there is unfinished business.
"If the money makes sense in the UK I would love to do it there. Deontay has never had a problem with travelling. That's not the issue at all."
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