But at least for now, Souza isn't taking the bait.
Branch (22-4) has accepted an offer to fight Souza (25-6) at UFC Fight Night on Sept. 22 in Sao Paulo, Brazil, but Souza has informed the UFC he's not interested in that matchup.
That doesn't sit well with Branch, who points out he fought an unranked, dangerous contender in Thiago Santos in his last fight -- and believes Souza should be willing to do the same.
"Whether I get the fight or not, people need to understand and see this," Branch told ESPN. "This is part of your legacy and who you are. If a fighter has a dead right to fight you, you take it. Thiago Santos was on a four-knockout streak. There was nothing I could have said to dispute the fact he earned the opportunity to break into the top 10.
"I would not have been a real fighter if I had declined that fight. I would have been a chicken---- and the UFC would have probably cut my ass, rightfully so. You know what I'm saying? Real fighters fight each other."
Gilberto Faria, Souza's manager, confirmed to ESPN they declined the fight.
Faria says Souza, 38, is concentrated on a rematch against Luke Rockhold, Yoel Romero, Kelvin Gastelum or even the champion Robert Whittaker, all of whom have beat him before. Souza is coming off a split decision loss to Gastelum in May.
"Every fight Jacare has taken so far, his opponent has been ranked lower than him. We are looking to fight someone ahead," Faria said. "Jacare gave a rematch to Derek Brunson [earlier this year], who he beat before. Jacare has never gotten the opportunity of a rematch.
"Rockhold, Romero, Whittaker, Gastelum -- those were all great fights. Why not a rematch? Also, Chris Weidman would be a great match, which people have wanted to see. This time we've decided to sit and wait."
Branch, of Brooklyn, says he's not ready to give up on the Souza fight yet -- and even intends to personally send Souza a text message respectfully asking him to accept.
A former WSOF champion at middleweight and light heavyweight, Branch is 2-1 since moving to the UFC last year and has won 12 of his last 13 overall.
He says he's not trying to jump any lines with this callout, he's just asking for what he feels he's earned.
"I was asked to fight him two times before, but they were short notice and didn't make sense as a professional athlete. Right now, we both have the opportunity to train and it makes perfect sense.
"If I can't get this fight, then I'll look for something else -- but for me, in my career, almost 37 years old, fighting MMA for 11 damn years, it makes sense. Why do I need to argue for this? I'm a dedicated athlete and I want the fights that will reward me."