The Blues are believed to be on the lookout for a new striker despite signing Nicolas Jackson from Villarreal during the summer.
Jackson has struggled for consistency since his arrival, scoring six goals in 14 Premier League games, with three of those strikes taking place against nine-man Tottenham Hotspur in November.
Chelsea's other main centre-forward option, Armando Broja, is still finding his feet following a lengthy spell on the sidelines.
With the Blues looking to add further quality to their forward line, they are expected to delve into the transfer market in the new year.
A recent report suggested that the West London club had made Osimhen a priority target for the January transfer window.
According to FootballTransfers, Chelsea are willing to pay whatever is needed to bring the Nigeria international to Stamford Bridge.
The report claims that the Blues are determined to sign Osimhen and are prepared to break their transfer record, which was set in January this year with the €121m (£105.6m) signing of Enzo Fernandez from Benfica.
While Chelsea are ready to splash the cash, there is uncertainty about how they would pull off the deal within Financial Fair Play regulations.
Osihmen was the subject of two offensive TikTok posts from the club's official account earlier this season, while he also had a difficult relationship with former Napoli boss Rudi Garcia, who was replaced by Walter Mazzarri last month.
The Nigerian striker netted 26 goals in 32 league appearances last term to help end Napoli's 33-year wait for the Serie A title.
Although he has gone on to score six times in 11 league matches this season, Napoli are well off the pace set by Inter Milan and Juventus.