The Reds agreed a £5.1million deal with Red Star Belgrade before Christmas and the 19-year-old midfielder travelled to England at the weekend to finalise the move.
He will undergo all his physical examinations on Monday but no confirmation is expected until Tuesday at the earliest, after which the youngster will rejoin Red Star on loan until the end of the season.
Liverpool fought off late interest from Russian clubs CSKA Moscow and Zenit St Petersburg, who were offering over £7m, and Grujic chose Anfield after personal intervention from Klopp after being advised about the under-21 international by his Serbian assistant Zeljko Buvac.
The Reds are also tracking Schalke defender Joel Matip, who is out of contract at the end of the season, but there is still plenty of work to be done to sign the German-born Cameroon international.
However, despite their on-going defensive frailties any deal would be for Matip to join in the summer.
But Press Association Sport understands there are at least two French clubs who are ahead of Liverpool in the running for the 24-year-old while Schalke have not given up hope the centre-back will sign a new contract.