The striker was filmed partying at home with several guests, in breach of government regulations during the coronavirus pandemic.
"Everton Football Club was appalled to learn of an incident in which a first-team player ignored Government guidance and club policy in relation to the coronavirus crisis," the club said in a statement.
"The club has strongly expressed its disappointment to the player and made it clear that such actions are completely unacceptable.
"Everton has regularly stressed the importance of following all the Government guidelines -- including rules and advice for inside and outside of the home -- through a series of official communications to all staff members, including players."
Kean, who joined Everton last summer from Juventus, is one of several Premier League players who have breached lockdown regulations and are now likely to face disciplinary action.
Tottenham's Moussa Sissoko and Serge Aurier apologised after the pair were photographed training together, contrary to regulations.
Manager Jose Mourinho was also forced to apologise when he was pictured at an outdoor training session with midfielder Tanguy Ndombele.