Salah, who scored during the Anfield rout, can be seen scrolling through his phone in the car's driving seat in the clip, as shown by the Mirror.
Merseyside Police tweeted that they had been made aware of the incident and revealed it 'has been passed to the relevant department'.
ℹ️ We have been made aware of a video believed to show a footballer using a mobile phone whilst driving. This has been passed to the relevant department. Thanks for letting us know 👍— MerPolCC (@MerPolCC) August 13, 2018
In the video, crowds can be seen gathering around the car hoping to have items signed or photos taken, before Salah drives a few metres forward while appearing to be on his phone.
A Liverpool spokesperson told the Liverpool Echo: “The club, after discussion with the player, have made Merseyside Police aware of the footage and the circumstances surrounding its capture.
"We have spoken to the player also and will deal with any follow-up internally.
"Neither the club or player will be making any further comment on this matter.”
The video had been widely shared by a Facebook account but was deleted on Monday.
Salah is an icon on Merseyside, having scored 44 goals in 54 appearances during the 2017/18 season and leading the side to the Champions League final, and is seen as one of Jurgen Klopp's most important players.
The Egypt international scored Liverpool's first goal in the West Ham win, tapping in from close range before Sadio Mane netted a brace and Daniel Sturridge headed in late on.