Markovic spent last season on loan at Fenerbahce.
Fenerbahce's Lazar Markovic in action with Celtic's Kieran Tierney and Stefan Johansen
Markovic spent last season on loan at Fenerbahce in a move sanctioned by Klopp's predecessor Brendan Rodgers.
The Serbia international signed for the Reds from Benfica in the summer of 2014 but struggled to fit into the side - scoring just the three goals from 34 appearances in his first season on Merseyside - while there were times he found himself the first to be substituted.
Chelsea's Filipe Luis in action with Liverpool's Lazar Markovic
Last season wasn't the best of times for Markovic either as one or two injury problems prevented him from getting the kind of game time he would have liked. He played 21 games in all competitions for the Turkish side scoring just the two goals.
However, with an early return to pre-season training, one can't fault the attitude of the 22-year-old as he looks to show that he can have a future on Merseyside.
Given the side isn't in Europe, Markovic will face a fight to get in the side if he is to be kept at the club for the coming season.
Sadio Mane's signing will make Markovic's chances that little more difficult but given Liverpool lacked pace last season - something which Markovic has - Klopp will want to have a look at him to see if he can at least be an option for the side for the coming season. If he can demonstrate he can play in more than one position then that will only help his cause.
It will be interesting to see if his services are retained, or if Klopp decides it's best to cash in on the player in order to strengthen in other areas.
Liverpool manager Juergen Klopp during training