Tottenham are ready to bid £12m for Burnley striker Danny Ings, according to Sky sources.
Gareth Bale has given a strong indication that he expects to remain with Real Madrid next season.
The 23-year-old Norway international will go straight into the Hoops squad which travels to Poland on Tuesday morning for the first leg of their Champions League qualifier against Legia Warsaw.
Berget - who comes with an option to buy for the Scottish club – said: "I am delighted to join Celtic and I'm looking to make the very best of this great opportunity.
"I know what a magnificent club Celtic is and what the club means to the supporters - it is a real privilege to be here.
"I also know the manager very well and I am delighted to be teaming up with him again and, of course, the great squad of players at the club.
"I have been in the qualification (rounds) before but have never played in the Champions League so hopefully I will be able to get there.
"That is the goal for the club and it should be a goal for the club as big as it is."
Berget, who joined Cardiff City in January from Molde, spent a season on loan at Hoops manager Ronny Deila's former club Stromsgodset between 2010-11, during a three-year spell with Italian side Udinese.
Deila said: "I'm really pleased that we have been able to add Jo to our squad. I know his qualities and I am sure he can add something new to the team.
"He is a player who will work hard and give everything to the team. I hope he can enjoy a successful time at Celtic."