Grujic, 19, has been touted as the next great Serbian star having emerged through the ranks with Red Star Belgrade, and he even helped his country win the Under-20 World Cup over the summer.
Unsurprisingly, his exploits in Serbia have triggered plenty of interest in his signature, with the Daily Mail recently reporting that the likes of Celtic, Chelsea, Everton, Manchester City and Manchester United are looking to lure Grujic to Britain.
However, it appears that Liverpool have stolen a march on their rivals, as new manager Jurgen Klopp looks to add new signings to his squad at Anfield.
Grujic - who has earned comparisons to compatriot and Chelsea star Nemanja Matic - is quoted by the Daily Mirror as saying that he has held talks over a move to join Liverpool, but no agreement has been reached with the Reds.
The teenager also claims that he won't be moving in January, as he wants to try and help Red Star qualify for next season's Champions League, meaning a summer move may be on the cards.
"It is certain that I remain here until the end of qualifying for the Champions League," he said. "I had a talk with Liverpool, but I did not receive yet any agreement. I'm still a Red Star player."
"I'll try to respect the club, the fans and the general public, but I think that for myself I have to choose the best environment, where I will get the chance to play and where I work with good coaches in good conditions," he added.
The race for Grujic's signature may drag on for a while yet, but Liverpool are certainly in pole position right now having already held talks with the player - and Klopp will be hoping to fend off any competition for the teenager over the coming weeks and months.