England Under-21 international Ramsdale has also been linked with Aston Villa, but the Blades are the only club to have submitted a bid to the relegated Cherries.
If the deal is completed, 22-year-old Ramsdale will return to Bramall Lane, where he began his career.
He moved from United to the Vitality Stadium in 2017 for £800,000 and went on to play 37 times in the Premier League for Bournemouth, having made his debut against Chris Wilder's side last August.
Blades boss Wilder is in the market for a new keeper after loanee Dean Henderson returned to parent club Manchester United.
Bournemouth rejected an initial bid of around £12m from the South Yorkshire club for Ramsdale.
The Cherries originally hoped to hold on to the player, despite their five-year top-flight stay coming to an end.
But Stoke-born Ramsdale, voted Bournemouth supporters' player of the year for the 2019-20 campaign, is understood to be keen on the move to continue his career in the Premier League.
The £18.5m fee includes the 15 per cent sell-on clause United inserted into the initial deal between the two clubs.
Ramsdale is set to become the second player sold by the Cherries since relegation, following Nathan Ake's £40m switch to Manchester City.
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