The 22-year-old spent the second half of last season playing for the Bluebirds and appeared 14 times for Neil Warnock's side as Cardiff achieved top-flight football for the first time since 2014.
And while the team from South Wales have had a bid accepted by Liverpool for Grujic to have a second temporary stint at the Cardiff City Stadium, Galatasaray have also made an offer to the Serbian, with the Turkish side offering him the opportunity to play regular Champions League football.
According to Turkish newspaper Aksam, Grujic will make a decision next week regarding his playing destination next season.
The 22-year-old reported for pre-season training at Liverpool on Friday morning alongside superstars Mohammed Salah and Sadio Mané after the trio were given an extended break following their commitments at this summer's World Cup with their respective countries.
Grujic was part of Serbia's 23-man squad who travelled to Russia but did not feature for Mladen Krstajic's side as they were eliminated at the group stages.
Cardiff manager Warnock has been anxious for Grujic to return to Cardiff this season, but has been frustrated by his agent dragging out the transfer for as long as possible.
The 69-year-old told the BBC, "It's a waiting game, but we won't wait forever. I think his agent - as they do - probably thinks he is going to try and get more money elsewhere."
Warnock also mentioned that Jurgen Klopp prefers the idea of Grujic getting regular Premier League football for the Bluebirds, and that might influence the player to stay in the UK.
"Liverpool want him to come and I think it will be good for him" - he added. "I think Marco does as well."
Grujic joined Liverpool in January 2016 from Red Star Belgrade for a reported £5m fee but has only made 14 appearances for the Reds in that time.