The 24-year-old will be assessed at the club's Melwood training ground on Tuesday as manager Brendan Rodgers begins his assessment of the Molde centre-back.
Fulham and Belgian side Club Brugge were also linked with a move for Forren. His arrival, albeit on trial, leaves Liverpool defender Sebastian Coates' position in question.
Earlier this month Rodgers admitted the Uruguay international, who has made just nine appearances this season - only two in the Premier League as substitute, could go out on loan if adequate cover could be found.
"It has been difficult for him to get games and the best thing for Sebastian would be to go out and get his games," said the Reds boss.