The Championship side rejected a bid of around £12m last week and value the 22-year-old at closer to £25m. Ramsdale joined Bournemouth from Sheffield United for £1m three years ago.
Manager Chris Wilder wants a goalkeeper in place for the start of the new season. He had hoped to re-sign Dean Henderson on loan from Manchester United but they have still to make a decision on the 21-year-old’s future.
Sheffield United are lining up an improved £15million bid for Bournemouth’s Aaron Ramsdale
United may yet loan him out again but will look at other clubs and Wilder can’t risk waiting too long on their decision for such a key position.
Earlier on Saturday, Bournemouth announced that Jason Tindall will take over as manager after Eddie Howe left the Cherries following relegation from the Premier League.
Tindall was Howe’s assistant during his tenure as Bournemouth boss as amassed 199 appearances for the club during his playing career.
The Cherries rejected an offer from the Blades last week believed to be around £12million