The 25-year-old has been linked with a transfer to Anfield throughout the summer and looks likely to complete a £20million switch in the coming days.
Southampton paid Lyon £8.5m for the Croatia international 13 months ago and he had an impressive first season at St Mary’s.
Lovren’s exit will mean Saints have lost three key players from last season as well as manager Mauricio Pochettino.
Asked if he anticipates Lovren’s exit, Southampton boss Ronald Koeman said: “I expect everything.
“Because I know Southampton, I know the possibilities of the biggest club in England, and they had a great season, and it’s normal that you will lose some players.”
Adam Lallana and Rickie Lambert are already at Liverpool, while defender Luke Shaw sealed a move to Manchester United.
Southampton have already received over £50m on those sales, and could be about to take in even more cash with the sales of Lovren and Morgan Schneiderlin.
The French midfielder could be about to link up with former boss Pochettino at Tottenham in the next week.
Tottenham’s bitter rivals Arsenal are also interested in the player, who cost Southampton £1.2m when he arrived from Strasbourg in June 2008.