The Premier League champions have negotiated a guaranteed buy-back option, valid for three years, as well as a sell-on clause of 15 percent.
Chris Wilder had been determined to sign a forward in the window, with his club failing to score a league goal thus far this season, and the 20-year-old was his first option.
Brewster, yet to play a top-flight game, wasn’t registered in time to make his debut at Arsenal on Sunday.
The England youth international, who won the Golden Boot during the Under-17 World Cup triumph three years ago, starred at Swansea on loan last season, hitting 11 goals in 22 games across all competitions.
Brighton and Crystal Palace were among the clubs to show an interest in Brewster, but United’s pursuit was the most concrete and persistent.
Meanwhile, Xherdan Shaqiri is also set for a departure from Liverpool having been left out of the matchday squad for the Carabao Cup defeat to Arsenal on penalties on Thursday.
“He’s not injured and there were no disciplinary reasons,” Jurgen Klopp said.
“So he didn’t do anything wrong. It’s just the time of the year where sometimes the manager gets some information and then I have to react on that. We will see what happens.”
Liverpool are expected to trim their squad further, with the manager explaining: “I can work easily with 24 or 25 players over a long season, no problem, but if a player doesn’t feel comfortable, then we have to talk.
“That’s what’s going on in the moment.”