The Serbia international is one of several fringe players that Liverpool are prepared to let leave before the transfer window closes.
While the Reds have been open to bids, they are yet to receive an acceptable offer for Grujic, who spent the last two seasons on loan at Hertha Berlin.
As the transfer window nears an end, the 24-year-old is attracting interest from Liverpool’s Premier League rivals West Ham and Southampton.
And out of the two suitors, Ralph Hasenhuttl’s Saints are the keenest to land Grujic before the transfer window closes.
Southampton lost star midfielder Pierre-Emile Hojbjerg to Tottenham Hotspur earlier in the window and are yet to fill the void left by the Dane.
It remains to be seen if the Saints will turn their keen interest in Grujic into a formal offer in the final days of the window.
All eyes will also be on whether West Ham firm up their interest.
Liverpool are said to want a fee of £20m for the Serbian, who started in their EFL Cup defeat to Arsenal on Thursday.
