The 22-year-old joined the Reds from Red Star Belgrade in 2016, and has made 14 first-team appearances for the English giants.
Grujic spent the second half of last season on loan at Cardiff City, making 14 appearances for the Welsh outfit as they secured a return to the Premier League.
The Serbian turned down the chance to return to Cardiff this summer, however, and it is understood that he has also rejected the advances of Middlesbrough.
According to the London Evening Standard, a move abroad is likely, with Russian giants CSKA Moscow currently leading the race to sign the midfielder on loan for the 2018-19 campaign.
Grujic was not involved in Liverpool's Premier League opener with West Ham United on Sunday afternoon.