The Daily Star says the Reds have secured a deal with the Scottish side to bring the teenager to Merseyside.
Gallacher, 18, will be a part of Liverpool's Under-23 squad as he continues his development.
The Scotland Under-17 international spent a week training with the Reds earlier this month and Kop boss Jurgen Klopp has given the deal the green light.
Gallacher is understood to have attracted interest from the likes of Manchester United and Stoke.
But it is Liverpool who have ultimately won the race for the youngster.