Supporters' ire has turned on Everton's hierarchy in the last few weeks as they face a second successive season battling to retain their Premier League status.
It is believed Everton chairman Bill Kenwright last week received death threats via email, while chief executive Denise Barrett-Baxendale was grabbed by a fan and placed in a headlock as she left the directors' box after a recent game.
All board members were instructed to stay away from Goodison Park on Saturday, when thousands of supporters joined a post-match sit-in to protest against the running of the club.
Police also had to cordon off the exit to the players' car park, but it did not prevent the likes of Yerry Mina and Anthony Gordon being confronted in their vehicles on roads surrounding the ground.
Merseyside Police confirmed "no threats or incidents were reported to police prior to the game", but talks are ongoing to determine whether any offences have occurred.
A statement said: "We can confirm that Merseyside Police is liaising with Everton Football Club following alleged threats towards directors ahead of the Premier League fixture against Southampton on Saturday, 14 January.
"No threats or incidents were reported to police prior to the game, but we are in communication with the club to establish if any offences have taken place, and to ensure that any future reports are received through existing channels.
"Any threats reported to Merseyside Police would be assessed and investigated as a matter of course, and any appropriate safety measures implemented.
"We are also aware of videos in circulation of fans approaching players' cars as they made their way from Goodison Park after the match.
"At this stage no offences have been reported."
Everton slid to a 2-1 home defeat to bottom-of-the-table Southampton, the only team below the Toffees in the standings at the halfway point of the season.