Grujic has spent the last two seasons on loan in Germany at Hertha Berlin, where he made an impact and grew his reputation in the Bundesliga.
Liverpool are willing to let him move on if the deal is right and Grujic is attracting attention from Gladbach.
The west German club are admirers of his talents and it is claimed he could soon move to Germany.
Grujic made a total of 54 appearances across all competitions at Hertha Berlin and got on the scoresheet on nine occasions.
Jurgen Klopp included him on the bench for Liverpool’s Community Shield defeat against Manchester City in the season curtain raiser.
Liverpool have since brought in another midfielder in the shape of Thiago Alcantara though.
And if Grujic stays at Anfield beyond the closure of the transfer window next month he could find playing time hard to come by.