Per the source, Remy would fall further down the pecking order behind Michy Batshuayi and Diego Costa if Lukaku were to complete his move.
Remy struggled for playing time last season and could struggle further for action if he were to stay.
Chelsea have already made it clear they do not want to loan Remy and would prefer to sell him on a permanent deal.
Besiktas have been closely linked with the French striker but had a loan deal rejected by the Premier League outfit.
The Black Eagles have rekindled intersted in Remy according to Lig Tv and are hoping the Blues accept a cut-price deal for the striker considering he will be deemed surplus to requirements.
Besiktas won the league title last season scoring the most goals in the league but are currently without a first-choice striker after Mario Gomez decided he would not be returning to the club following the end of his loan stay.
The Istanbul giants will play Champions League football next season.