A short clip circulating on the internet appears to show Vieira lashing out at the fan, who was goading him, before some other fans helped the Palace boss to safety.
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LondonWorld understands that the Premier League is studying the clip and will investigate the incident.
The fan’s footage has now emerged, which shows him screaming abuse at the Palace manager.
“I have nothing to say about that,” Vieira told reporters when asked about the incident during his post-match news conference.
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Lampard, who also saw the incident involving Vieira, had sympathy for the Palace manager even though he did not entirely condemn the pitch invasion from his fans.
The Everton manager said: “I feel for Patrick. I didn’t get him at the end.
“I would have said, ‘come in with us’ - although he might not have wanted that.
“Of course he ran 80 yards across the pitch and it’s not easy.
“It was pure elation of fans who have come on the pitch after avoiding relegation.
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“If it is done in the right way, let them stay on the pitch, let them have their moment. As long as they behave, no problem.”
Patrick Vieira’s incident at Goodison comes at a time where there are growing concerns over pitch invasions in English football.
Earlier in the day, a Nottingham Forest fan was jailed after headbutting Sheffield United captain Billy Sharp after a pitch invasion.