Aston Villa are keen to offload Coutinho this summer as they seek to shed his substantial pay packet from their wage bill.
Real Betis were keen on Coutinho, but could not get a deal over the line and he is now looking at countries where the transfer window remains open for a Villa exit.
Qatari side Al Duhail want Coutinho and have put a lucrative offer to the Brazilian, but Besiktas do not feel they are out of the race.
They are intensifying their efforts to sign him and, according to Turkish daily Milliyet, are banking on the European factor being the difference maker.
Besiktas know that their financial offer to Coutinho is lower than Al Duhail’s.
A move to Besiktas though would keep Coutinho in Europe and also hand him the chance to play in the Europa Conference League this season.
The transfer window in Turkey remains open until 15th September, giving Besiktas time to do a deal.