Brewster, who helped England win the Under-17 World Cup last year, was signed from Chelsea by Liverpool during Brendan Rodgers’ tenure and is viewed as a hot prospect at Anfield, and has already been included in first team squads.
It is understood that negotiations are at an advanced stage, with both the Reds and Brewster happy for the move to go through, although other clubs remain interested.
Despite interest from other clubs, Anfield boss Jurgen Klopp is believed to be happy for Brewster to join the Scottish Premiership champions to gain more first team experience.
Brewster has only just turned 18, and hit an impressive 20 goals in 22 appearances for England Under-17s, including two hat-tricks in the Under-17 World Cup and the opening goal in the 5-2 win over Spain in the final.
Among his team mates were Manchester City’s Phil Foden, who has previously been linked with Celtic; highly regarded Chelsea midfielder Callum Hudson-Odoi and Borussia Dortmund youngster Jadon Sancho.
While he is yet to make an appearance for the Reds first team in a competitive fixture, Brewster has appeared in friendly matches for Klopp’s side.
With the future of both Moussa Dembele and on-loan Paris Saint-Germain striker Odsonne Edouard unclear, Rodgers is understood to be looking at bolstering his strikeforce.
The Hoops have been credited with an interest in Rubin Kazan’s Iranian forward Sardar Azmoun, Schalke striker Franco di Santo and Dnipro hitman Artem Dovbyk in recent months.
Edouard has expressed an interest in staying at Celtic on a permanent basis and the club have an option to buy the Frenchman as part of the loan arrangement with PSG.