Arsenal say they have identified an image of the person who threw a bottle at Tottenham's Dele Alli on Wednesday and are "embarrassed" by the incident.
England midfielder Alli, 22, was struck on the head by a plastic bottle during Spurs' 2-0 win in the Carabao Cup quarter-final at Emirates Stadium.
Police are investigating the incident.
"Anyone identified will receive a lengthy club ban and their details will be passed to the police to commence legal proceedings," said the club.
Alli was hit by a bottle thrown from the crowd as Arsenal prepared to take a throw-in in the 73rd minute.
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The Metropolitan Police says it is working with Arsenal to try to identify the person responsible, but no arrests have been made.
"We have all been embarrassed by the individual who threw a bottle at Dele Alli," Arsenal added in a statement.
"Behaviour of this type has no place at Emirates Stadium and after analysing CCTV footage, which showed him leaving the stadium after throwing the bottle, we have identified an image of the suspect.
"We are liaising with the Metropolitan Police and investigations continue in order to apprehend the culprit."
The Football Association is aware of the incident and will support the police and clubs as they look into the matter.
Alli will face no action for responding with a 2-0 gesture to fans - referencing the scoreline.
Pochettino applauds Alli's behaviour
Spurs manager Mauricio Pochettino praised Alli for the restraint he showed after being struck by the object thrown from the crowd.
"It was an amazing reaction," Pochettino said in his news conference to preview Sunday's match with Everton.
"In another country the player was going to be down on the pitch. Dele behaved really well. Arsenal should be grateful to him because the player could have gone down and created a massive problem.
"I think the behaviour was top from Dele. Sometimes he is criticised but he was very mature."
Arsenal's statement added: "We are not responsible for the actions of one individual, but send our apologies to Dele Alli and everyone at Tottenham Hotspur for this incident.
"We do not tolerate any anti-social, discriminatory or violent behaviour at Arsenal Football Club."
Pochettino added that Arsenal should not be punished for the actions of one person, claiming: "One person cannot create a mess in a club like Arsenal.
"Arsenal is going to take a big decision, with responsibility to fix. People have to behave and celebrate and support your team but in the right way.
"It's such a dangerous thing, if the bottle hit him on his head or his eyes we're then talking about a very bad thing.
"There's no point when you go to enjoy a game and that happens."
Alli had earlier scored the second goal and Spurs went on to reach the semi-finals with a 2-0 victory.
Wednesday's incident follows a banana skin being thrown towards Arsenal striker Pierre-Emerick Aubameyang in the Premier League match between the sides at the same venue on 2 December.
The Spurs fan responsible was fined and banned from football for four years on Tuesday.
BBC Radio 5 live pundit Dion Dublin, who was co-commentating on the game, said: "It's sad to see. It's mindless. Why would you do that?
"It must have had water or something in it to reach Dele Alli. Why would you risk being idiots?
"Just support your club, don't do ridiculous things like that."
When asked about the incident after the game, Alli told Sky Sports: "It is what it is. It made the goal a bit sweeter and the win."
An EFL spokesperson told BBC Sport the matter would be up to the Football Association to investigate, but they would "assist" if required.