Berisha, a full Norwegian international, has earned nine caps for his country and is valued at around £2.5m.
Meanwhile Rodgers has also been granted the funds by the club's hierarchy to launch an £8m offer for FC Basel winger Mohamed Salah after weeks of continued interest.
Salah has been at the top of Rodgers' agenda since last summer and Reds scouts have been monitoring the 21-year-old for much of the season.
It is thought that Rodgers is also keen to add a left-back, centre-back, midfielder and striker to his squad although he may now only make a move for Salah after the club informed him that funds would be low until the end of the season.