The Napoli striker has been heavily linked with a switch to the Blues in recent months amid his sensational form in Italy.
Osimhen scored 26 goals en route to Serie A title success last season, with another five coming in the Champions League.
He has already netted on eight occasions in the current campaign, and Napoli are doing all they can to keep hold of him.
The Nigerian international signed a new deal last week until 2026, with his release clause reported to be around £112million.
Despite committing his future to the club, Osimhen has given Chelsea fans cause for optimism after a reply he sent on social media.
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One supporter wished him a happy birthday in the early hours of Friday morning with an old picture of the striker in a Chelsea shirt.
"Pls come and rescue Chelsea soon," they pleased, with Osimhen then replying with two hearts and what appears to be a praying emoji.
It may seem subtle, but Blues fans on X have loved what they've seen as one said: "Chelsea blue is your colour mate. Come soon now, please! We're waiting for you."
"Just woke up my whole street at 3:53am," a clearly excited second added in response.
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A third simply asked: "There is a chance.. Please, Chelsea."
"Woke up to this. Man of man if we manage to sign him in January we're finishing in top 6. Bookmark this. He and Nkunku will cook together," said an optimistic fourth.
John Obi Mikel plays agent as he urges Napoli star Victor Osimhen to join Chelsea on his podcast
Osimhen's links with Chelsea are nothing new, with former Blue John Obi Mikel recently speaking to him on his podcast.
"You are 100 per cent going to Chelsea next season," he told him as the west London side eye a fresh attacking signing.
Mauricio Pochettino's side have failed to replace prolific forwards such as Didier Drogba and Diego Costa in recent seasons.
Nicolas Jackson signed in the summer and despite scoring as many goals as Osimhen this season, still appears to struggle in front of goal.
Chelsea fans have finally seen Christopher Nkunku play following his lengthy spell out with injury and he has already impressed.
The potential signing of Osimhen, whenever it may be, could be the key to unlocking the true potential of this Blues side.