Costa underwent surgery on his left biceps in early October, his manager, Wallid Ismail, told ESPN.
UFC president Dana White first revealed the surgery news on the YouTube show "The Schmo and The Pro."
According to Ismail, Costa suffered the injury in his victory over Yoel Romero in August. He initially injured the biceps in March 2018.
Ismail said Costa was hoping to avoid surgery, but after his trip to Melbourne, Australia, in early October to watch Adesanya's title bout against Robert Whittaker, which Adesanya ultimately won, doctors advised Costa to have surgery.
Costa's team is hopeful he'll be able to fight in April.
The problem, according to Ismail, is the UFC had penciled the Adesanya-Costa fight for March. According to sources, the promotion is now looking at new options for Adesanya's first title defense.
"I have just heard that UFC will find another opponent for Adesanya," Ismail told ESPN. "I can't even imagine Adesanya fighting against anyone else other than Paulo Costa. Paulo will be ready to fight in April 2020, and this is the kind of fight all MMA fans want to watch. They truly hate each other, and they are both undefeated."
Costa (13-0) is 5-0 in the UFC, while Adesanya (18-0) is 7-0. Should they meet with perfect records, it would mark just the second male title fight in UFC history involving two undefeated fighters.
There is no love lost between the two fighters, who have taken verbal shots at each other over the past few months.
"This fight has all the ingredients to break all UFC pay-per-view records," Ismail said. "Paulo is certain Adesanya is going to lose any time soon, but he really wants to be the first to beat him up."