The 28-year-old signed for Mourinho’s side during the summer from Stoke.
For large parts of the season he has found himself second choice behind Thibaut Courtois.
Liverpool were initially linked with Begovic whilst he was still at Stoke, but the move to Chelsea happened and it is thought Klopp’s side may re-ignite their interest in the new year.
It is thought The Reds also have AC Milan ‘keeper Diego Lopez in their sights as another possible option.