The Nigeria international faces an uncertain future at the Stadio Diego Armando Maradona following controversy created by Napoli's TikTok account.
The club's official account came under fire after posting two posts which appeared to mock Osimhen, with one making light of his penalty miss against Bologna, while the other compared him to a coconut.
In response to the now-deleted posts, Osimhen's agent threatened to take legal action against the reigning Serie A champions.
Napoli insisted that no offence was intended with the two videos, but they stopped short of offering an apology.
While talks have previously taken place over a new contract, there is now considerable doubt as to whether Osimhen will sign a new deal with the club.
The 24-year-old, whose current contract runs until 2025, has attracted interest from a number of clubs, including Chelsea.
Napoli club president Aurelio de Laurentiis has stated publicly that he is confident Osimhen will put pen to paper on a new deal.
However, according to Calcio Mercato, Napoli have started to draw up contingency plans in case Osimhen leaves the club in January or at the end of the season.
The report claims that Jesus has been identified as a potential target alongside Atletico Madrid's Alvaro Morata and Lille's Jonathan David.
Jesus has been hampered by injury since joining the Gunners from Premier League rivals Manchester City in the summer of 2022.
Despite being affected by injury issues, the Brazilian has still managed to make 41 competitive appearances for Mikel Arteta's side, registering 14 goals and eight assists.
Morata, who previously played in Italy for Juventus, has made an impressive start to the season, scoring seven goals in eight games for Atletico.
David, meanwhile, has found the net four times in 11 games this term, and the Lille striker is also a target for AC Milan as well as Napoli.