It all started when Stallone arrived late to his table and found there was no seat there for his wife, actress Jennifer Flavin.
There are differing accounts as to how other people at the table -- including Manchester by the Sea star Affleck and producer Matt Damon -- treated Stallone before he and Flavin left to watch the show and their three Miss Golden Globe daughters from backstage.
"He was understandably upset," one E! News source said of Stallone, while a second witness said Affleck and the other stars at his table attempted to make more room for an additional chair.
But People magazine said it talked to someone who said Stallone was embarrassed because no one at the table made an effort to accommodate him and his wife.
"There weren't enough seats and so there was confusion, but it wasn't personal against him," another person told People.
E! said Affleck later approached Stallone backstage and tried to make things right with him, but Stallone wasn't in the mood to talk.
"Casey was nothing but gracious," an insider said. "Stallone was horrible to him and told him to get out of his face."
"The story you refer to is an interesting rumor -- but a rumor nonetheless. Sylvester Stallone had a wonderful evening attending and presenting at the Golden Globes and celebrating with his family and friends," his publicist told People, without addressing the details regarding the alleged awkwardness with Affleck.
Affleck's representative declined to comment.
Affleck ended up winning the Best Actor in a Drama prize, while Stallone and his Rocky co-star Carl Weathers presented the Best Drama award to Moonlight.