Damon who shares a comedic, on-air feud with Kimmel, made his presence known after Clooney teased that he was ready for his 4-month-old twins Ella and Alexander whom he shares with wife Amal, to make their television debut.
Damon then showed up pushing a double stroller while wearing a backpack and a blanket.
"I'm his manny, it's a nanny who's a man," Damon said to Kimmel. "You're not a man," his on-screen rival said in response to which Damon replied, "That's not what your wife said."
Kimmel continued the war of words going by asking, "So based on your physique I guess you're breast-feeding the children too?"
"No, no, I actually just wanted them to get their nap, so I bought them to a place where they wouldn't be woken up by laughter," Damon said.
As it came time to unveil the twins, Damon went into the stroller and instead pulled out two middle fingers.
Clooney and Damon were on hand to promote their latest film, Suburbicon which arrives in theaters on Oct. 27.
On Monday, the pair appeared on Good Morning America together to discuss the ongoing Harvey Weinstein sexual assault scandal in Hollywood and how they were unaware of the extent of the producers sexual misconduct.