The Turkish club had a reported 10million euro (£8.3m) bid accepted by Inter president Massimo Moratti this week, with Galatasaray director Bulent Tulun requesting a prompt response from the player.
Former Denmark midfielder Soren Lerby, who represents the 28-year-old, had previously declined to comment on any negotiations but broke his silence today.
Quoted in the Gazzetta dello Sport, Lerby said: "The talks started yesterday and today a delegation from the club arrived to meet him.
"Now the decision lies with him, anything is possible in football and Turkey is fast becoming a hot destination.
"If everything goes to plan the deal will go through. We will wake up tomorrow and everything will be more clear."
Sneijder has cut an isolated figure at the San Siro this season, making just five league appearances and publicly refusing to accept a two million euro wage cut.
Turkish Super Lig leaders Galatasaray are on a three-week winter break and eager to add Sneijder to their ranks before next week's trip to Kasimpasa.
Gala director Tulun told Turkish radio station Radyospor yesterday: "Sneijder will give us a response inside 24 hours, but in 99 per cent of these cases the transfer negotiations are already done."
Sneijder has reportedly attracted the interest of Manchester United, Manchester City, Liverpool and Tottenham, as well as Ligue 1 outfit Paris St Germain.