The pair exchanged words as Fury's promoter Frank Warren said the bout was "official" and added more details would be made public in the next week, BBC Sport reported.
Wilder said Fury would "experience" being knocked out for the first time.
Fury, 30, won all 10 rounds to secure a points’ win over Pianeta in Belfast.
"One thing I promise when I go to Las Vegas is I am knocking you out," he told Wilder, 32.
The American, who was holding his belt as the pair took turns on the microphone at Windsor Park, responded, "He has never had the WBC belt. If he even thinks about that belt, he better wake up and apologize to me as he will never have this."
The prospect of a bout between the two seemed close in January 2016 when they clashed in the ring following Wilder's victory over Poland's Artur Szpilka.
But Fury did not fight again until June this year and taking on such a formidable opponent so quickly after his time away seemed unlikely until talks gathered momentum in recent weeks.
Fury, who showed sharp movement and focus throughout his win over Pianeta, has secured the fight after drawn-out – and ultimately unsuccessful – negotiations between Wilder and WBA, IBF and WBO champion Anthony Joshua.
"We have two men who will fight anyone," added Fury. "This man has been trying to make a fight with another chump.
"They called, I answered. I said: 'Send me the contract.' They sent it. I said 'yes'."
Warren has not yet confirmed Las Vegas as the fight's venue, but boosting pay-per-view interest will now fall into focus, and two of the sport's most vocal characters will need little help in delivering sound bites.
There have been warm words exchanged in the past, with Wilder praising Fury for battling mental health issues and the Briton insisting his rival deserved more from negotiations with Joshua.
But, when the talking is done, Fury will need all of the evasive footwork he showed against Pianeta and more against a man with 39 knockouts from 40 bouts.