Bjornebye is one of the older players in the squad, having been brought in during Graeme Souness' reign. His best season came in 1996/97 when he was possibly the club's most consistent performer. Bjornebye excels in an attacking wingback position and at his best, he plays a part in many of the Reds attacking initiatives. He has shown what he is capable of by making it into 96/97's Premiership's Team of the Year, but has had difficulty playing up to that standard since then.

With so many younger players in the side, Bjornebye looks to be out-of-favour with Gerard Houllier. The Norwegian has spent the 1999/2000 season as captain of the high-flying reserve side. His attitude has been examplary and he has not had any misgivings since being dropped to the reserves. There are a few Norwegian clubs interested in signing the former international, although he has not expressed any desire to leave the club, it looks certain that he no longer has a part to play in the first team.