The online footage, which is believed to have been taken following the Reds' 4-0 win against West Ham United on Sunday, shows the player sitting in traffic and then driving off.
Salah can be seen scrolling through his phone while sat in the driver's seat, with a number of supporters gathered around his vehicle.
Liverpool say that they are aware of the footage and are in contact with the police in an attempt to resolve the matter.
A club statement read: "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."
Merseyside Police posted on Twitter: "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."
The footage has since been taken down from the Facebook account which had posted it.