The move comes just a week after captain Giovanni Di Lorenzo agreed to extend his own contract until 2030, with Manna now turning his attention to Anguissa, whose current deal has just nine months left to run.
According to Tuttosport, discussions over a new agreement have been ongoing for several weeks, with Napoli offering a contract through to 2029 while the Cameroon international is understood to be pushing for terms extending to at least 2030.
Negotiations are expected to continue with further contact planned before the upcoming international break, as both sides look to find common ground on the length of any extension.
Anguissa's current contract, signed after his permanent move from Fulham in 2022, runs until the summer of 2027, and Napoli are keen to secure his long-term future given his importance to the midfield under head coach Antonio Conte.
The contract talks come at a busy moment for Napoli, who are still digesting a recent defeat to Inter Milan and preparing for their first Champions League match of the season against Arsenal at the Maradona on Wednesday.
The club have prioritised tying down key players in recent weeks, with Di Lorenzo's new deal completed last week and the Anguissa situation now the priority as Manna works to secure the futures of the squad's senior figures before turning his full attention to the winter transfer window.
Anguissa, 30, joined Napoli from Fulham in 2022 following an initial loan spell, and has since become one of the club's most consistent performers, playing a key role in the Serie A title win under Antonio Conte.
He has previously been linked with interest from several clubs, including sides in Ligue 1, the Premier League, Turkey and the Middle East, but has shown little inclination to leave Naples, with reports in Italy suggesting he turned down multiple approaches during recent transfer windows in order to remain part of Conte's project.
Securing his long-term future alongside that of Di Lorenzo would represent a significant statement of continuity for a club looking to build on last season's title success while also competing on multiple fronts this year.