Rapaport posted screenshots of the pair's Instagram direct messages on Twitter and Instagram in which Durant appeared to use homophobic, misogynistic and profane language.
"I receive threats and disgusting messages DAILY, but never in my wildest dreams did I think [Kevin Durant] would be among them," Rapaport wrote on Tuesday afternoon.
"The [snake emoji] himself is now threatening me, bringing up my wife and wants to fight. This is supposed to be America's sweetheart right?"
According to the screenshots, the confrontation began last year when Rapaport criticized Durant's post-game interview after the Nets' season-opening victory over the Golden State Warriors in December.