Police and the Football Association are investigating after Crystal Palace boss Patrick Vieira was involved in an altercation with a fan following their Premier League defeat at Everton.
Television footage showed him being taunted by an Everton supporter as fans ran on to the pitch at full-time.
The Frenchman appeared to kick out at the fan before being ushered away by other supporters.
Police say they have not received a formal complaint about the incident.
“Merseyside Police is investigating an altercation on the pitch at Goodison Park following the Everton v Crystal Palace match on Thursday,” the force said in a statement.
“We are working with Everton to gather all available CCTV footage and are speaking to witnesses.
“No formal complaint has been received and enquiries into the incident are ongoing.”
Everton staged a stunning comeback to win and retain their top-flight status.
“I have nothing to say about that,” Vieira said when asked about the incident during his post-match news conference.
Everton manager Frank Lampard 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.”
On the pitch invasion, Lampard added: “It was pure elation of fans who have come on the pitch after avoiding relegation.
“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.”
Speaking on Sky Sports, former England striker Dion Dublin said: “We know how overjoyed the Everton fans are but you can’t be doing that. You can’t be pushing managers and pushing players, touching players and shouting in their faces.
“We know it is only one or two who spoiled it for everyone else. Having fans on the pitch is not the way forward. We need to stop letting fans, or allowing fans to get on the pitch. It is too dangerous.”
Vieira’s confrontation at Goodison Park comes amid rising concerns over pitch invasions in English football following a number of recent high-profile incidents.
Footage on social media also appeared to show Swindon players involved in a confrontation with Port Vale fans following Vale’s League Two semi-final shootout win on Thursday.
The other League Two semi-final, played on Wednesday, also ended in a pitch invasion with Mansfield boss Nigel Clough saying such incidents are “getting concerning” after his player Jordan Bowery appeared to be shoved by a fan late on in their win at Northampton.
Earlier on Thursday, a fan was jailed after running on to a pitch and headbutting Sheffield United captain Billy Sharp at the end of the Blades’ Championship play-off semi-final shootout defeat by Nottingham Forest on Tuesday.