A statement on Everton's website said: "We have identified one individual whom action will now be taken against.
"This will include a ban from attending future Everton fixtures and the Club registering a formal complaint with Merseyside Police."
The incident happened in the 63rd minute of Thursday's Europa League match after Everton's Ashley Williams barged into Lyon keeper Anthony Lopes.
Players from both sides ran into the melee as the pushing and shoving ended up directly in front of the stands.
A man in a tracksuit, holding a child with one arm, can be seen in TV pictures striding to the front before apparently throwing a punch at Lopes.
The man continues to shout and gesture angrily as Lyon players confront him.
Several other men from the crowd were also involved in the disturbance, with one grabbing the arm of Lyon's Lucas Tousart.
Merseyside Police said: "Detectives will be reviewing footage and speaking to witnesses to establish what, if any, offences have been committed."
Williams, Everton's captain for the night, was booked, as was Lyon's Bertrand Traore, but European football bosses could also review the skirmish and take action.
"Ashley Williams and Everton will surely face further sanction from UEFA once video evidence is studied," former Premier League referee Graham Poll told the Daily Mail.
"Williams totally lost the plot.
"There were numerous players from both teams involved... and even a fan from the Everton section at the Gwladys Street End became involved - first pushing then appearing to aim a punch at a Lyon player."
He added: "The UEFA delegate and referee's observer will have met with the official after the game and discussed their findings before submitting individual reports.
"Unless the referee ill-advisedly says he saw everything, sanctions really must follow."
Speaking after Everton's 2-1 home defeat, Williams told BT Sport: "It's what happens, it's football. It is what it is.
"It's high emotions. We want to win the game, they want to win the game and stuff happens. That was just one of those times."