Chelsea, Arsenal and Manchester United are all thought to be admirers of Osimhen, but Napoli's valuation of the forward - thought to be €100m (£84m) - has made a switch to the Premier League unlikely.
PSG are seemingly firmly at the head of the queue for the 25-year-old, as the French giants bid to bring in a replacement for Kylian Mbappe following his move to Real Madrid.
According to SPORT Mediaset, Osimhen's agent has flown to Paris to speak with the Ligue 1 champions, and Luis Enrique's side are becoming 'increasingly confident' of securing his signature.
PSG are now said to be expecting to close the deal in the near future.
Osimhen has scored 76 goals and registered 18 assists in 133 appearances for Napoli in all competitions, including 17 goals and four assists in 32 outings during the 2023-24 campaign.
Reus 'closing in' on MLS move
Meanwhile, according to transfer expert Fabrizio Romano, Los Angeles Galaxy are closing in on the signature of ex-Borussia Dortmund attacker Marco Reus.
The 35-year-old scored nine goals and registered 10 assists in 42 appearances for Dortmund during the 2023-24 campaign, but he ended his long association with the club on June 30.
There has been a host of speculation surrounding Reus's future in recent weeks, with a number of clubs believed to have expressed an interest in securing his signature.
Romano, though, claims that LA Galaxy are set to sign the forward on a free transfer.
Reus scored 170 goals and registered 131 assists in 429 appearances for Dortmund in all competitions.
The MLS outfit are said to be now finalising personal terms with the German, and confirmation of the move is expected to arrive in the near future.
Birmingham 'register interest' in Barry
Elsewhere, Birmingham City have allegedly expressed an interest in signing Aston Villa striker Louie Barry during this summer's transfer window.
The 21-year-old made the move to Villa from Barcelona in January 2020, but he has only managed one first-team appearance for his current side.
Barry has recently spent time out on loan with Ipswich Town, Swindon Town, MK Dons, Salford City and Stockport County, scoring nine times for the latter in League Two last season.
According to Football Insider, Birmingham have now been in touch with their rivals Villa to discuss the possibility of signing the forward during the summer market.
The report claims that Villa are open to allowing Barry to leave, as he is not believed to be in the plans of first-team manager Unai Emery during the 2024-25 campaign.
Birmingham will play in League One next season following their relegation from the Championship.