Southampton have tried to sign the 26-year-old centre-back in the last two transfer windows, but have kept up their pursuit in the belief that the Italy international wants to move to the Premier League.
Astori is set to report for duty with the Azzurri at the Confederations Cup later this month and would prefer his future to be cleared up before the start of the tournament in 11 days' time.
Cagliari are hoping that the widespread interest in Astori forces his price up above the £15m mark and believe that £10m could be a starting point for negotiations with two years left on his contract.
AC Milan boss Massimiliano Allegri is understood to have put Astori on a three-man shopping list of Italy internationals with a move to the San Siro meaning Astori would join a club who co-owned him for a year earlier in his career.
But it is Spurs' interest that could be the biggest threat to Southampton's chances of luring Astori to St Mary's with a report in Tuttosport claiming the Londoners could be prepared to go as high as £15m for him.