Berget has held talks with the Premier League club and is on the verge of joining countrymen Magnus Wolff Eikrem and Mats Moller Daehli at the Cardiff City Stadium.
The 23-year-old, who can also play in attack, played under Solskjaer at Molde and has opted not to sign a new deal with the Norwegian club.
Cardiff manager Solskjaer confirmed holding discussions with Berget, but said on Friday: "No other additions have been concluded."
Berget started his career at Lyn and had a spell in Italy with Udinese before returning to his homeland.
He has been capped by Norway four times and now looks set to join Cardiff’s battle to stay in the top flight.
Wilfried Zaha is reportedly set to join Cardiff on loan, but Solskjaer denied the Manchester United winger has undergone a medical.
Cardiff are also said to be closing in on Wigan defender Ivan Ramis.
Solskjaer's side travel to Championship side Bolton Wanderers in the fourth round of the FA Cup on Saturday and the former Norway striker sees the tie as an excellent opportunity to secure a morale-boosting win.
"We're changing things gradually here. It will take time, but tomorrow will be an important game," he added.
"An FA Cup run would be vital for us to build momentum and a good feeling around the camp.
"We've got to build a belief that we fully belong in the Premier League and can compete at grounds like the Etihad (Stadium) and Old Trafford."
‘Arteta, Alonso, Emery, me… none of us were physical players – we needed the understanding of the game. That probably helped us move into management’: Premier League boss reveals reasons for natural career progression
‘England have the players to win the World Cup – it’ll be tough for Thomas Tuchel to do a bad job, with the squad he has at his disposal’ Former Three Lions winger backs new boss after gentle qualifying draw
