The Turkish giants are understood to be putting together a loan deal for the 20-year-old, who has struggled to make an impact since his move from Barcelona in a deal worth £8.6million.
Spurs paid £4.7million up front for Giovani but he could not break into the starting XI when Harry Redknapp took over from Juande Ramos as manager and he was loaned to Ipswich for the final part of last season.
During the current campaign he has made three appearances and his progress has been hindered by an ankle injury suffered in the Carling Cup win over Preston.
Galatasaray have been busy in the January market in England, having also signed Lucas Neill from Everton and taken Brazilian striker Jo on loan from Manchester City. The Turkish side also signed Brazilian playmaker Elano from City in the summer.
One player who has returned to White Hart Lane from a loan spell is John Bostock, with the player Spurs signed controversially from Crystal Palace struggling to get in the Brentford team.
"He's got terrific ability, you just hope he will do it," Redknapp said.
"He can do it. Maybe I need to get him working with and training with the first XI, get him in the squad a bit and see what he can do."
Source: PA
Source: PA