WBC heavyweight champion Wilder has long been talking with Anthony Joshua about a unification bout but, even with Wembley Stadium secured for the Briton for April, no contract has been agreed.
Eddie Hearn, Joshua's promoter, touted a fight between Wilder and Dillian Whyte at the weekend, too, but it now appears that the American will instead face Fury.
Fury (26-0) ended a two-and-a-half-year absence by beating Sefer Seferi in June and faces Francesco Pianeta in August, before turning his attentions to Wilder (40-0).
"I can confirm that me and Deontay Wilder are in negotiations, very close to being done," Fury said in a video message on his Twitter page on Monday.
"It is out there now, and I can reveal that negotiations have been very strong for December. We're almost done for this deal.
"I won't be looking over Francesco Pianeta. He's going to give me the right fight and the right work to prepare me for Deontay Wilder.
"I've got to flatten him first and then, Wilder, let's dance. You've been dealing with s***houses in the past, with Eddie Hearn and Joshua, but I am a man of my word.
DOWN TO BUSINESS
— Frank Warren (@frankwarren_tv) July 29, 2018
"If I say I'll fight you, I'll fight you. End of.
"Prepare well, train hard, let's get the deal done. I'm going to knock Francesco Pianeta into next week and then I'm going to knock you spark out, too. Good luck."
Frank Warren, Fury's promoter, had earlier confirmed his intention to seal a date with Wilder.
"[Whyte and Dereck Chisora] are two competitive and very good fighters but, with due respect to them, we're actually trying to make the fight with Deontay Wilder," he told talkSPORT.
A second video update from Fury suggested the bout would take place in New York City or Las Vegas, before claiming he later plans to fight in Africa and, humourously, Antarctica.
"I can say I'm looking to fight in Antarctica, because I believe you guys need a big event out there," he said. "Who bigger and better than the Gypsy King to come to Antarctica and put on a show?
"Get ready; the Fury Express is coming somewhere near you."