Reports in Italy suggest that Roma had a £5m bid for Astori dismissed out of hand earlier this week.
United have been tracking Astori since last summer, while Tottenham are also thought to hold a long-standing interest in the former Milan player.
Manchester City, Everton and Southampton have all shown interest in Astori in the past but it is thought that United and Tottenham have the best chance of getting the Italian international to quit Serie A.
Indeed, while Cagliari are resigned to losing Astori this summer after fighting off interest for the past two years.
But Massimo Cellino – who remains chairman at the club despite also becoming a majority shareholder at Leeds United earlier this month – is not willing to let his prize asset go cheaply.
In July 2012 Astori himself turned down a move to Russian side Spartak Moscow after Cagliari accepted a £12m offer for the player, but it seems their asking price may now have dropped.
It now remains to be seen whether United are ready to pursue their interest in Astori without a degree of certainty over who will take over from David Moyes in the long term, with Tottenham facing a similar issue in regard to Tim Sherwood.
Quite how Leeds fans would feel about Cellino agreeing to sell Astori to rivals United as he targets a return to the Premier League before the 2015-16 season, also remains to be seen.
Written by Steve Milne