The 20-year-old frontman still has four and a half years left in his deal at the Let 1 club.
However, Sky Sports claims that the Premier League is monitoring their progress between the three teams hosting them as they plot a move.
Nigerian international Osimane has made headlines with his performance in the French top flight this season.
The former Wolfsburg ace is currently the second-highest-performing striker in the division according to Wyscout statistics.
The stats are only above Monaco’s Wissam Ben Yedder, with stars like Kian Mapppe and Memphis Depp.
He is expected to remain in Lille for the rest of the campaign – until he gets an offer next month which is too good to refuse.
And Chelsea may be preparing for a splash of cash to help raise Frank’s qualification expectations for the Champions League due to the recent slump for the men.
Olivier Giroud is expected to leave Stamford Bridge in January, and after winning their battle to obtain their two-window transfer restrictions, they could move on to increase their striking department in the coming weeks.
Osman, who has scored 12 goals in 23 matches so far this season, won one while performing against the Blues in the Champions League.
It is believed that he was not in a hurry to leave Lily, currently ranked third in Liz 1, and is happy to continue developing in his game.
However, Jose Mourinho’s Spurs could put his London neighbours and current love leaders, Liverpool, in line for his signature.
Johan Sacramento, now No. 2 at Tottenham, worked with the forward at Lille before joining the Spurs.
And Lille’s sports director Luis Campos, who signed Ossimen from Charleroi, Belgium last summer, is also considered a close friend of Mourinho.