Media playback is not supported on this device
The Spaniard was sent off following the clash with the Everton midfielder in the 84th minute at Goodison Park.
A Chelsea spokesman said: "Diego spoke to club officials and expressed regret over his reaction to the challenge from Barry that led to his red card.
"But Diego was also very clear that he did not bite him at any point during that altercation."
Chelsea manager Guus Hiddink claims Everton deliberately provoked Costa, who had a running battle with Barry throughout the tie.
In the incident that led to Costa's second yellow card, the 27-year-old confronted Barry, moving his head towards him and in the direction of his neck.
"He was chased a bit in the game by Everton. They went after him. They knew it. It is within the rules," said Hiddink.
"As a referee you have to protect the situation, but knowing and feeling this atmosphere."
Hiddink, who managed Chelsea to victory over Everton in the 2009 FA Cup final, said he had not seen the incident that led to Costa's dismissal.
"I try to be fair in my judgement and it is difficult for me to say yes or no so I don't want to give judgement on this," he said.
Media playback is not supported on this device
Everton manager Roberto Martinez said midfielder Barry, 35, had not complained about the incident - and was more interested in a win that takes his side to a semi-final at Wembley rather than any incident involving Costa.
Martinez said: "My interpretation is I don't think it was a key moment. It was an emotional game and rightly so.
"Diego Costa has a fighting spirit and I would like to praise the referee. The sending off of Diego Costa was right as I thought it was a second yellow card and the sending off of Gareth Barry was right.
"After we have won a game like this and got to Wembley, which our fans deserve so much, the last thing I am going to do is see if an opposing player has bit my player. Gareth Barry has said it is nothing to worry about. He is just disappointed he got a second yellow card."
It was Costa's first sending off since he joined Chelsea in summer 2014, although he received a retrospective three-game ban in January 2015 for stamping on Liverpool's Emre Can in a League Cup semi-final at Stamford Bridge.
The 27-year-old was also punished retrospectively with a three-match suspension after he clashed with Arsenal's Laurent Koscielny in a Premier League game at Stamford Bridge in September 2015.
Hiddink was asked whether Chelsea had considered an anger management cause for the confrontational Spain striker and joked: "There are movies about that, aren't there? Wasn't Jack Nicholson in a film called Anger Management? Maybe we can go and watch it together."
Former Premier League referee Howard Webb told BBC Radio 5 live: "I've watched the incident back and I am pretty confident that [referee] Michael Oliver has shown Costa a second yellow card for adopting an aggressive attitude.
"There is no way Michael Oliver can see what happens then with the neck - if it is a bite or not. He couldn't see it anyway as he is looking at the other side of Gareth Barry.
"Two things I think will happen now. I think the FA will look at the footage so they can try and see if a bite has taken place.
"They will look for evidence, for Gareth Barry's reaction, for marks on the neck. They will then come to a judgement and that will go down as something unseen by Michael Oliver.
"Secondly, I'll be amazed if Costa is not punished for his reaction after the second yellow card. He fails to leave the field of play, he is aggressive again to Michael Oliver and referees are always told to report that situation. That can lead to another match ban.
"The only time you could be sent off for attempting to bite is if you make the action to bite and the other player pulls away.
"But it looks like he has thought about it but not gone through with it. From what I've seen of the footage, it is not sufficient to support a charge for violent conduct retrospectively."
Media playback is not supported on this device
Striker Diego Costa has denied biting Gareth Barry during Chelsea's 2-0 FA Cup quarter-final defeat at Everton.