Earlier this week, it was reported that the Cherries rejected a bid of around £12m for the shotstopper, who kept five clean sheets in the 2019-20 Premier League season.
With Dean Henderson looking set to return to parent club Manchester United ahead of next season, Chris Wilder is understood to be searching for a new man to start in between the sticks for the Blades.
According to the Daily Mail, Sheffield United will increase their offer for Ramsdale to £15m, although Bournemouth value the goalkeeper at £25m and are expected to rebuff the Blades once again.
Having suffered relegation to the Championship at the end of the most recent campaign, Bournemouth, who recently appointed Jason Tindall as their new manager, will likely be resigned to losing a number of key players this summer.
Former defender Nathan Ake has joined Manchester City in a £41m deal, and speculation continues to grow over the future of Ramsdale, who still has just under four years left to run on his current deal with Bournemouth.