Per the source, Lucas has agreed terms with the Lions and will travel to Turkey to put pen to paper.
NTV Spor report that Galatasaray have offered €3 million for the services of the 29-year-old.
There are however, conflicting reports from England claiming that Klopp will try to block Lucas leaving the club unless defensive cover is found.
Klopp is currently facing a defensive crisis with Mamadou Sakho, Joe Gomez and Joel Matip all out injured and Skrtel as well as Toure recently parting ways with the club.
The Echo report that Klopp will only give the green light for Lucas to leave should Liverpool bring in a replacement.
If Klopp were to veto the move due to the injury crisis he is facing in defence it would be an almost groundhog day like situation for Lucas.
The Brazilian midfielder was on the verge of joining Besiktas last summer before Brendan Rodgers pulled the plug on the deal due to injuries in his squad.
Lucas would end a nine-year stint at the club if he were to leave and join former teammate Martin Skrtel in Turkey this summer.