Jurgen Klopp is already looking to boost his Reds squad ahead of next season, despite sitting at the top of the Premier League and in a three-way title race with Manchester City and Tottenham.
According to La Repubblica, the Reds placed the offer for Insigne through agent Mino Raiola and his father - as the Reds didn't want to approach Napoli chairman Aurelio De Laurentiis due to his volatile past.
The offer was almost immediately rejected by De Laurentiis when the news filtered through the outfit.
The fee was around half of the price tag the Serie A side have slapped on their prized possession - with the Italian open to offers of £131m.