Liverpool had invited Forren to England for a trial, but Southampton appear to have moved ahead of Brendan Rodgers's side in the race for the 24-year-old.
Forren's agent is reported to be meeting representatives from the Premier League strugglers in London on Tuesday in the hope of agreeing personal terms.
It is understood that Southampton are hopeful of completing the signing of the defender, who has five Norway caps, in the next 24 hours.
Molde manager Ole Gunnar Solskjaer has admitted that the Norwegian club had accepted a bid for the player, who has also been linked with Everton and Reading.
"There have been many inquiries from various English clubs, but Southampton submitted an offer," the former Manchester United striker told Norwegian newspaper Romsdals Budstikke.
"It was very interesting for us and Vegard. We are more or less agreed with the club, but nothing is a hundred percent confirmed.
"Molde is close to a deal, so we'll see if this is something that all parties will go with."
Southampton manager Nigel Adkins, however, has refused to confirm whether a fee has been agreed.
"Like everything through the transfer window, there is always going to speculation and people talking about different deals and I think it is prudent that, until someone signs a bit of paper, you don't announce it.
"Already in this transfer window in January there has been one or two cases where a player is meant to be at one club and then the next minute they have signed for another club.
"Speculation is there and obviously this is the latest one and when things happen, if they happen, our football club will announce it."