Liverpool were credited with serious interest in Insigne towards the end of last month by Italian newspaper Corriere dello Sport.
The 27-year-old has scored 10 goals and provided six assists in 26 Serie A appearances under Carlo Ancelotti this season.
And the playmaker would represent a valuable asset if Klopp was to add him to his squad of two-time Champions League finalists.
Klopp played down Liverpool’s interest in Insigne, suggesting the fact he’s tied down at Napoli until 2022 would be to big an issue to navigate.
However, he could be tempted to rethink his stance on Insigne after Napoli president De Laurentiis admitted his captain could leave.
“I understand that he has a contract, he is a Napoli player and he is from Naples,” De Laurentiis told Tutto Mercato Web, when asked if Insigne will be at Napoli next season.
“He has always shown great attachment to the shirt, now he is also a captain and has great responsibilities.
“However, this responsibility has played in the past, and could play him in the future, an idiosyncrasy with the fans that led to misunderstandings, I can’t force him to stay.
“Several times four years ago he asked me to leave because he didn’t feel loved.
“He asked me this even before the clarification meeting with [agent Mino] Raiola, but then we also talked about lengthening the contract.”
De Laurentiis’ comments on Insigne’s future come after Jurgen Klopp ruled out Liverpool making a move for him.
“We will not sign Lorenzo Insigne,” Klopp said. “He is a very good player but he has a long-term contract at Napoli. We will not even try it.”
Klopp has form for changing his mind, however, having signed Alex Oxlade-Chamberlain from Arsenal after saying he would not.
Klopp also critiqued Manchester United’s mega spend on Paul Pogba, before splashing out on Virgil van Dijk and Alisson.
Liverpool are now a attractive proposition for the world’s best players again having reached two Champions League finals and challenged Manchester City for the Premier League title this season.
Klopp will be eager for them to bolster their reputation when they face Tottenham in European football’s biggest club game on June 1.