Inaki Williams gave Athletic, who beat Real Madrid in the semifinal, the lead in the 94th minute, and with time running out and Barca down 3-2, Messi was sent off for lashing out at Villalibre in frustration.
The Royal Spanish Football Federation's competition committee will meet this week to decide the length of the Barca captain's suspension.
Messi, 33, is certain to miss two matches but that could be doubled, depending on how the competition committee categorise the dismissal. The ban will be served across La Liga and the Copa del Rey.
Therefore, he will miss Thursday's Copa trip to third-division side Cornella and next weekend's game against Elche in La Liga. If the ban is longer, he could also miss the next round of the Copa if Barca progress and the match against Athletic at Camp Nou on Jan. 31.
Sunday's red card was the first Messi has received while playing for Barca's first team, a length of 753 matches. It was the fourth of his career, though, following one dismissal for Barca B in 2005 and two with Argentina, including one on his international debut that same year.
Koeman said after Sunday's loss that he empathised with Messi's frustration.
"I can understand what Messi did," Koeman said. "I don't know how many times they fouled him, and it's normal to react when they keep trying to foul you as a player who is looking to dribble with the ball, but I need to see it again properly."
Barca went into Sunday's final on the back of a nine-game unbeaten run in all competitions and boosted by Messi passing a late fitness test after missing the semifinal win over Real Sociedad.
Two Griezmann goals looked set to extend that run to 10 matches, but the Frenchman's strikes were cancelled out by first Oscar De Marcos and then Villalibre before Williams struck a brilliant winner.
"I am pissed off, angry, annoyed," Griezmann told reporters after the defeat, which denied him his first piece of silverware since joining Barca from Atletico Madrid in 2019.
"I have all the bad feelings you can have after losing a final. We made mistakes at set plays, we defended poorly, we didn't speak at free-kicks. They're important details."
Koeman tried to put a brave face on despite the loss.
"We're sad and disappointed but I still think we're on the right path and we're improving," Koeman added.
"We have to improve defensively. We must be more committed. We can't let players turn, shoot and pass. But I am giving everything. I'm not the worst coach now [after a defeat] just like I am not the best after a win, but it's a tough one to take."
For Athletic, it was a first trophy since they beat Barca to win the Spanish Supercopa in 2015 and it came in just the third game of new coach Marcelino's tenure.
"The credit goes to the players," Marcelino said. "They deserved to be here for what they did last season and to win these two games [against Madrid and Barca] is incredible.
"I'd also like to remember Aritz Aduriz and Mikel San Jose and the other players that were here last season but are no longer part of the squad. And, also, the previous coach [Gaizka Garitano] and his staff."