The 25-year-old Croatia defender, who played in all of his side's matches at the World Cup, has spoken publicly about his wish to join the Reds.
Lovren will undergo his medical at Anfield before Liverpool return from their tour of the United States.
Saints boss Ronald Koeman said the club are "close" to signing a replacement.
Speaking after a 1-0 win over Bournemouth in a pre-season friendly on Friday, the Dutchman said: "Our job if someone leaves is to be prepared for that position.
"Because Lovren is a key player in Southampton we are working hard to bring someone in and are close to doing that.
"The club are working hard and it is still not too late."
Liverpool refused to comment.
Striker Rickie Lambert, 32, was signed for £4m from Southampton
Midfielder Adam Lallana, 25, was signed for £25m from Southampton
Midfielder Emre Can, 20, was signed for £10m from Bayer Leverkusen
Winger Lazar Markovic, 20, was signed for £20m from Benfica
Liverpool have already bought Rickie Lambert and Adam Lallana from Southampton this summer.
Lovren, who joined Southampton from Lyon for £8.5m last summer, will leave St Mary's 12 months into a four-year contract.
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