Ariel Helwani of ESPN confirmed the report and said that a location is to be determined for the fight, which is slated for Sept. 19.
Adesanya holds a perfect 19-0 record (8-0 UFC). Fourteen of his wins have come via knockout. His most recent fight occurred at UFC 248 in March against Yoel Romero, which ended with a unanimous-decision win.
The fourth-ranked pound-for-pound fighter in UFC's rankings will take on the No. 2 middleweight contender in Costa, who is also undefeated (13-0) and coming off a unanimous-decision win over Romero.
Mike Bohn of MMA Junkie confirmed that both fighters' camps have agreed to a fight.
The UFC 253 card will include bouts between Mackenzie Dern and Randa Markos, Hakeem Dawodu and Zubaira Tukhugov and Shamil Abdurakhimov vs. Ciryl Gane, per Bohn.
But the main event should have plenty of fireworks, and the lead-in might be epic. As Helwani noted, the two have exchanged a "war of words" since last year.
Of note, Costa's manager, Wallid Ismail, told Cole Shelton of BJPenn.com that Adesanya had rejected fights against Costa for UFC 251 and 252.
"UFC offered Adesanya to fight on July 11 and August 15 and Adesanya declined both times," Ismail said. His coach claims Adesanya hasn't gotten an offer which is not true. We have proof."
The two fighters will be facing off in two months, however, as one of the more anticipated MMA matchups of the year is set to take place.