The Turkish sports newspaper report that the Lions have offered Liverpool €4 million to sign the Brazilian international and are waiting for a response.
Per the source, Galatasaray have been locked in talks with Lucas for the past few weeks and are nearing a deal.
Lucas would end a nine-season spell at Liverpool were he to make the move to the Turkish giants.
Galatasaray have been in pursuit of a central midfielder since Felipe Melo joined Inter Milan on transfer deadline day last summer.
The Istanbul giants will be able to fully focus on the league campaign next season as they have been issued a one-year European competition from from UEFA for failing to meet FFP regulations.
Galatasaray could face competition from Besiktas who have also been closely linked with Lucas and almost signed him on a one-year loan last season.
Lucas did pick up an injury in the Reds friendly against Huddersfield last week and the club are waiting for results regarding how long he will be out for.