Brewster has been the subject of interest from a number of Premier League clubs in the transfer window, with the Blades and Crystal Palace at the forefront.
Sheffield United manager Chris Wilder is a huge admirer of the youngster and the Bramall Lane club have been chasing the player for two months.
The Blades have now made an initial offer of £18m, which could rise up to £23m after add-ons, in an attempt to land Brewster from Liverpool.
The Yorkshire-based club have been trying to get a deal over the line for the striker and are set for a big boost as it is claimed Liverpool are ready to accept the bid.
Jurgen Klopp rates Brewster highly, but Liverpool have been forced to look to cash in on the England Under-21s star in an attempt to balance the books, having completed the big-money signings of Thiago Alcantara and Diogo Jota.
Crystal Palace still want to sign Brewster and the striker is interested in the prospect of turning out for the Eagles.
Roy Hodgson’s men though are not willing to include a buy-back clause in the agreement, which is something Liverpool favour.
Sheffield United, however, have agreed to the Merseyside-based club’s demand, aiding them in their efforts to sign the player.