The Southampton based newspaper says that the Croatian defender will not turn up for pre-season training and has already handed in a transfer request.
The paper says: "Liverpool want to sign Lovren and it is understood that at least two approaches from the Reds have been knocked back this summer.
"The defender, who has also been linked with Tottenham, is unhappy at the stance of the Saints board, and believes an agreement was in place with him that if another club offered a certain amount, believed to be around £20m, he would be allowed to leave.
"Liverpool are said to have matched that figure, only for Saints to stand firm and knock back the offer."
Lovren agreed a four-year deal with Southampton last season when he signed for £8.5m from French side Lyon.
The 24-year-old was member of the Croatian team that failed to get out of Group A at the World Cup in Brazil.
Liverpool are expected to have plenty of money to spend with Luis Suarez set for a big move to Barcelona.
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