The 25-year-old Italy international is said to be valued at £12million by his club, with Southampton ready to better Liverpools bid of £7million by offering £9.5million to bring him to St. Mary’s.
The Saints are currently 18th in the Premier League, only out of the relegation zone on goal difference, and boss Nigel Adkins would hope that signing Astori wold help sure up their leaky defence and aid their push for survival.
The defender who started his career at Milan has been capped twice by Italy and was included in the squad which was beaten by England in a friendly last year, while also being considered for their Euro 2012 squad.
Failure to land Astore would come as a blow to Liverpool chief Brendan Rodgers who is keen to bolster his squad during the winter transfer window.
So far the Reds have acquired the services of forward Daniel Sturridge from Chelsea, but have struggled to iron out a deal for young winger Thomas Ince from Blackpool, who was allowed to leave Anfield just 18 months ago.
Astore would provide good competition for places for first choice centre back pairing Daniel Agger and Martin Skrtel, with current back up Jamie Carragher now 34-years-old, and Sebastian Coates relatively inexperienced.