Critchley spent a number of years coaching the 21-year-old in the Liverpool academy set-up before taking the head coach role at Blackpool.
The Seasiders visit Bramall Lane this afternoon with three points and three places between the sides after 14 games.
United paid Liverpool £23m for Brewster in October last year but the youngster has found it tough going, scoring just once in 38 appearances for the Blades.
His value sky-rocketed in the previous season as he netted 10 times in 20 games on loan at Swansea City and although he has not been able to replicate those returns with the Blades, Critchley believes Brewster will start to show his quality soon.
"I coached him for quite a few years in Liverpool so I know him well,” he told The Gazette.
“He’s a great boy to work with and I’ll be looking forward to seeing him and speaking to him. His best quality is scoring goals, he’s a natural goal-scorer. Hopefully he doesn’t hit the back of the net against us.
"I just think he’s going through a period of his career many young players go through, where not everyone takes the elevator to the top easily. Sometimes you have to take the stairs.
“It might be a little bit up and down and there’s very, very few players, like a Trent Alexander-Arnold [Liverpool right-back], where they break into the team and stay there.
“A lot of young players now have to find a different pathway, they have to go out on loan and it might take them a little bit longer.
“There’s no doubting Rhian’s qualities, both as a player and a person. I fully expect him at some point to show the quality he’s got, I just hope it’s not on Saturday.”