The Lions had initially made a loan offer but the report claims that the Reds want a permanent transfer fee as Lucas has just a year remaining on his contract.
The Lions will use goalkeeping coach Claudio Taffarel as an intermediary to convince Lucas into making the move if Liverpool accept a £3 million transfer fee.
Galatasaray could however, face competition from Super Lig champions and Istanbul rivals Besiktas who are also reported to be in the race.
Lucas was reported to be open to the prospect of joining Besiktas at the start of the month according to the Daily Star.
The Brazilian midfielder was closely linked with a loan move to Besiktas but the deal fell through at the last minute when former Reds boss Brendan Rodgers decided to keep the defensive midfielder following injury to Jordan Henderson.
“It’s been very intense [the window], to be honest,” he told Liverpoolfc.com.
“I think the last couple of years, every transfer window my name is [linked with a move].
“Of course there are times when you think you might have to move on and I was very close to moving but one thing I will always have is commitment to this club.”
Galatasaray will not play European football next season as the club are currently serving a UEFA competitions ban for failing to meet FFP regulations.
Besiktas meanwhile will play Champions League football next season after winning the league title.