Liverpool looked to be leading the race for Forren after they invited the Norwegian international over for a trial this week.
However, the trial has now been cancelled after Southampton made a firm bid for the highly-rated central defender.
Solskjaer has confirmed the two clubs have all but agreed a deal and Forren is now set to hold talks with Southampton about a possible move to St Mary's.
"There have been many inquiries from various English clubs, but Southampton submitted an offer, it was very interesting for us and Vegard," Solskjaer told Romsdals Budstikke.
"We are more or less agreed with the club, but nothing is a hundred per cent confirmed.
"Molde is close to a deal, so we'll see if this is something that all the three parties will go on."
However, Southampton could face competition for Forren's signature with Queens Park Rangers also thought to be weighing up a move for the 24-year-old.
QPR boss Harry Redknapp is in the market for defensive reinforcements with Ryan Nelsen set to leave for Toronto FC and he could make a move for Forren.