The report claims that the Lions are considering putting forward an offer to the Uruguayan international’s agent.
Ramirez is currently available on a free transfer as his contract is set to run out at the end of the month.
Southampton are yet to renew his deal and Ramirez is currently on international duty with Uruguay at the Copa America 2016.
Ramirez spent last season on loan at Middlesbrough scoring seven times and providing four assists in 18 games.
The 25-year-old joined Southampton on an £11.4 million move from from Bologna in 2012 but failed to settle in at the club and has spent two seasons on loan at Hull City and Middlesbrough.
Galatasaray would be unable to offer Ramirez European football this season as the club is currently suspended for one season from European competition for failing to meet UEFA FFP regulations last season.