The Spain international was sent off late after clashing with Barry, who was also sent off moments later for his second booking after fouling Cesc Fabregas.
Television replays suggest Costa went to bite his opponent before thinking better of it at the last second.
When contacted by Press Association Sport, the Football Association said it was aware of the incident and would wait for referee Michael Oliver's report before a decision would be made about any possible action.
Chelsea later issued a statement on the club's official Twitter account maintaining the forward had not bitten Barry, but did regret his reaction.
Interim Chelsea manager Guus Hiddink said in his post-match conference he would reserve judgement on the incident until he has had chance to see it for himself.
"I try to be fair in my judgement, I haven't seen it and it is difficult for me to say yes or no," said the Dutchman, who has not yet spoken to his player about it.
"Without seeing, I don't want to give judgement on this.
"He was chased a bit in the game. They went after him. They knew it. It is within the rules.
"As a referee you must protect the situation but knowing and feeling the atmosphere - then comes the moment that I would like to give judgement (on) but I haven't seen it."
Hiddink was asked whether the volatile Costa should seek help for anger management and jokingly made reference to the 2003 film of the same name.
"There are movies about that, I think. One movie with Jack Nicholson. Maybe we go and watch it together," he added.
"I like him still going towards the edge but you also have to take into consideration he was desperate to play today.
"It would have been better to rest him but it is not the type of game you rest players.
"It is all the combination; not fully physically fit and all the provocations and he has to take responsibility for the incident."
Everton manager Roberto Martinez, who was grateful to two moments of brilliance from striker Romelu Lukaku before the red cards to settle an otherwise disappointing tie, played down the incident.
"My interpretation is I don't think it was a key moment. It was an emotional game and rightly so," said the Catalan, who had not spoken to Barry after the game.
"Diego Costa had a fighting spirit and I would like to praise the referee (Oliver). The sending-off of Costa was right.
"The sending-off of Gareth Barry was right and the referee never allowed those emotions to stop the fluency.
"Whatever happens with Diego Costa I am sure the two players shook it off.
"From where it was he moved his head towards Gareth Barry and I lost sight of that action.
"I thought it was a second yellow. It was a moment that didn't have an impact on the scoreline."
Martinez rejected the suggestion his players had targeted the temperamental Costa.
"It is part of trying to do an interpretation as you want," he added.
"We matched each other well. None of my players fell into the focus of losing the game."