That bid was flatly rejected by Liverpool, who value the 22-year-old at around double that amount. The Reds are also keen for the former Red Star Belgrade man to pen a new contract at Anfield before they consider a loan move, with interest mounting from across Europe.
Grujic turned down the chance to return to Premier League new-boys Cardiff City, where he spent the second half of last season. He made 14 appearances for the South Wales club, and is keen to secure regular first-team football again this season.
He will have no shortage of suitors, with the likes of Lazio and AC Milan having been strongly linked this summer. Grujic told reporters last month that he would leave any decision over his future to manager Jurgen Klopp, but it looks likely that he will depart Anfield, at least temporarily, before the month is out.
Despite that, Goal understands that he remains highly rated by Klopp and his coaching team, who see huge potential in the 6'3" prospect. Grujic has already made 14 senior appearances for Liverpool, including three starts.
The Reds expect to offload a number of other players to European clubs before the end of August, with Simon Mignolet, Lazar Markovic, Pedro Chirivella and Divock Origi all in line for moves. Offers for defender Ragnar Klavan, meanwhile, will be considered.