Right-back Randall, 20, is set to be rewarded for his progress into Jurgen Klopp’s first team while keeper Fulton, 19, has impressed at under-21 level.
The Liverpool Echo report that Randall will pen terms on a long-term deal while Reds coaches are confident Fulton can progress to senior level.
Randall, who joined Liverpool’s academy aged six, was handed his debut by Klopp in October’s Capital One Cup win at Bournemouth and started the 6-1 win at Southampton in the same competition.
He is likely to be given another opportunity in Thursday’s Europa League clash against FC Sion with the Reds already through to the last 32 stage.
Fulton has been involved in the match-day squad twice and is currently third-choice behind Simon Mignolet and Adam Bogdan.
The Burnley-born stopper has been with Liverpool since under-10 level.