I don't know, Klopp is not a miracle worker. The problem with Markovic so far has been that he couldn't adjust to the different level of intensity compared to what he was used to in Portugal, if anything Klopp's tactics will require even more intensity.
Klopp will probably play him in his natural position and not as a wing back.
He also might continue playing him when he's had a good game.
Wheter or not he makes it is a different question, but he'll have a better chance.
There was some talk of Markovic wanting to leave so I'm guessing the relationship with Rodgers wasnt the best.