Throughout the summer transfer window, the Nigeria international was heavily linked with a switch from Napoli to either Chelsea or the Saudi Pro League.
However, after both of those options fell by the wayside, it was Galatasaray who swooped to secure a season-long loan deal for the 25-year-old.
Osimhen has since contributed six goals and four assists from his eight appearances for the Turkish giants, including a double in a Europa League victory over Tottenham Hotspur on Thursday night.
That performance showed the level that Osimhen was capable of producing against Premier League opposition, a potential boost for Chelsea supporters ahead of their club making a renewed attempt to sign one of world football's most highly-regarded frontmen.
What has Osimhen said on Chelsea?
Nevertheless, it appears that Osimhen has different ideas for the remainder of his season having pledged his short-term future to his current club.
Speaking after the win over Spurs, as quoted by The Standard, Osimhen said: "I'm here as a Galatasaray player, I'm happy here. Since I came the Galatasaray fans, the club, I saw how big a family this club is.
"The way they have treated me and my family in a wonderful manner. It's important to be able to give them my all on the pitch.
"I've seen a lot of the speculation linking me out of the club in January, I can state through me now I'm going to be here until the end of the season."
A blow for Chelsea?
Chelsea have recently committed to handing Nicolas Jackson a nine-year contract, a clear show of faith in a player who has continued to improve and impress in the Premier League.
If Osimhen were to make the switch to the West Londoners, it would cast doubt as to where Jackson would feature in the team given that Osimhen would realistically only move to Chelsea if he was regarded as first choice.
Enzo Maresca's side have struggled for goals in recent games against top-flight opposition, yet that does not necessarily mean that they would or should jump at the chance to sign Osimhen.
Christopher Nkunku, Joao Felix and Marc Guiu are all backup options to Jackson, who would become an alternative on the left flank should another centre-forward be signed to compete for his place in the attack.