White told ESPN's Brett Okamoto the contracts haven't been signed but the plan calls for Adesanya, who won the first meeting by split decision in April 2018, and Vettori to headline the June 12 event.
Adesanya, 31, is set for a return to the middleweight division after a failed attempt to become a dual champion. He took on Jan Blachowicz for the light heavyweight title at UFC 259 in March, but he lost by unanimous decision.
"The Last Stylebender" is still the king of the middleweights, however, earning four straight wins in title fights in the 185-pound division. Most recently, he successfully defended his belt with a second-round TKO of Paulo Costa in September.
"You'll see me back at 205 later in the future, but right now I've got a division I've got to dominate," Adesanya told reporters after the loss to Blachowicz. "I know they're probably thinking, 'Oh, that's it. He just got taken down.' All right, cool. But I'll remind them again why I'm the king at 185."
His career MMA record still stands at an impressive 20-1 with 15 knockouts.
Meanwhile, Vettori has won all five of his bouts since the loss to Adesanya to improve to 17-4-1. He most recently beat Kevin Holland by unanimous decision April 10.
The 27-year-old Italian took a shot at the middleweight champ Monday on Twitter:
Vettori hung with Adesanya for three rounds in the first matchup, and he's showed progress over the past couple of years, so he represents a serious threat to the titleholder.
If he pulls off the upset, there would probably be an immediate rematch to complete the trilogy.