The Norwegian striker, who can also play behind the front man, will arrive at Parkhead initially on loan.
But the Hoops have an option to make the move permanent, if the 23-year-old shows the form which has won him four caps for his country.
Ronny Deila worked with Berget for a short time at Stromsgodset six years ago, when he went back to Norway on loan from Udinese.
Berget moved to Cardiff in January from Molde. Ole Gunnar Solksjaer went back to his previous club to make the youngster one of a group of signings he made during his first transfers in charge of the Welsh club.
However, faced by a host of other strikers vying for a game, Berget only made two appearances for Cardiff as they battled in vain to avoid relegation from the Premier League.
Now he hopes to use the platform Celtic are offering him to kick start his career.