The Blades have been tracking the 20-year-old for some time and the PA news agency understands he has travelled to South Yorkshire for his medical.
As part of the deal Liverpool have negotiated a 15 per cent sell-on clause and a buy-back option which will be active for the next three seasons.
Brewster scored 11 goals in 22 appearances on loan at SkyBet Championship side Swansea after joining them in January.
He played just four times in cup competitions for Jurgen Klopp’s side, his last appearance coming in the Community Shield against Arsenal at the end of August, with his missed penalty in the shoot-out giving the Gunners victory.
While highly-rated at Anfield – hence the buyback option inserted into the Sheffield United deal – his pathway to the first team has been hindered by the fact their front three of Mohamed Salah, Roberto Firmino and Sadio Mane play virtually every week and Divock Origi and new signing Diogo Jota remain ahead of him in the pecking order.
Blades manager Chris Wilder remained tight-lipped over the potential deal during his pre-match press conference.
Wilder said: “I can’t confirm that it’s all been done but we have been in negotiations with Liverpool for a while and he has been on our radar for a while and discussions between the player and the club are still ongoing.
We are always trying to bring better players into the football club and ones that can improve the group
“There’s nothing to announce at this minute but we are down the line.
“Obviously when it happens I can confirm that but as we are at this particular time we are not over the line. I don’t think you can guarantee anything until it gets over the line.
“We are always trying to bring better players into the football club and ones that can improve the group, and it is an area we need to improve, and that’s not being disrespectful to the strikers we already have here, we need competition in the group.
“We need different options for different games. I’ve said consistently that we need to bring someone in and we have worked hard to identify players. Bringing better players into the group will improve us and help bring us better results.”
