The revelation comes despite the club being forced to publicly apologise for “tapping up” the Southampton defender.
The Saints have not withdrawn their initial complaint after they alleged Liverpool made an illegal approach to try and sign the Dutch international.
But the Premier League are understood not to be considering further action over the matter unless new information is provided.
Premier League chiefs have been reluctant to get involved and wanted the clubs to resolve the matter privately with Liverpool offering both a private and public apology.
It was revealed that Jurgen Klopp sent Van Dijk a string of messages and Liverpool met with the player to try and tempt him to Anfield.
There are strict Premier League rules about “tapping up” and Chelsea were hammered in the most high profile case after trying to sign Ashley Cole when he was still at Arsenal.