Although the Reds spent over £170million on reinforcements last summer, Klopp still feels there are key areas for improvements in his side as he seeks to land his first trophy.
The German tactician is in need of a new No 10, and according to Italian news outlet Rai Sport, Insigne has emerged as a target.
The report indicates that the Reds are already keeping tabs on the Napoli star, who has been transformed from a winger into a striker, ahead of next summer.
However, it is thought that the Naples outfit have no desire to sell the 27-year-old.
Insigne impressed against Liverpool when he scored the winner in a 1-0 victory for the Serie A side when the two sides met in the Champions League group stages earlier this month.
The pacey star was a constant menace that night and gave Trent Alexander-Arnold a torrid time.
Although it is not known how much Insigne would cost, Napoli wouldn’t let him leave cheaply.
Should Liverpool land him, it would keep Mohamed Salah on his toes.
After a blistering campaign last term, Salah has struggled and scored just three times in the Premier League this term.
When asked about Carlo Ancelotti’s tactical switch of his position at Napoli, Insigne said: “Talking to the coach, he had asked me to try out this new position,” Insigne said.
“I said I’d give it a go and I’m really enjoying it.
“The coach asked me to make fewer sacrifices in defence and focus more on being incisive in front of goal, which is what I’m doing.”
Liverpool, meanwhile, return to Premier League action away to Huddersfield on Saturday (5.30pm).
One positive bit of news is that James Milner is in contention to start after recovering from a hamstring injury that he sustained against Manchester City 10 days ago.
Although Salah and Virgil van Dijk also picked up knocks during the international break, they could be back in time.