Napoli would be looking to cash in on the 27-year-old, who has spent eight seasons at the club after rising through their youth academy early in his career.
Insigne broke a lot of Red hearts a couple of weeks ago in the Champions League when he went head to head against the Liverpool side who are on the verge of landing him in England in coming January.
The Italian left Liverpool youth sensation Trent Alexander Arnold for dead several times throughout the game and was proving to be a nightmare for the Reds and he got his reward for the same in stoppage time.
As Salah is looking a bit complacent as compared to last season and Firmino inconsistent with his performances, Klopp would love to have his options open by bringing Insigne to Anfield in January.