Kent has been on loan at the League One side for the first half of the season.
Kent appears to have impressed during his time at the Sky Blues having made 16 appearances for the League One side, and the report suggests the Reds would like to get a closer look the player, and may fast track him into the first team.
The 19-year-old can play out wide on both flanks and used in the 'number 10' role, which is highly competitive part of the field but there are plenty of injuries to the side at the moment that more options are required during a crucial period of fixtures for the club which could see them compete on four fronts.
The Liverpool boss is said to want more options available to him in the Europa League, which gets back underway in February when the Reds take on German side Augsburg in the last-32 of the competition.
With the current injuries mounting up, it is of little surprise that Klopp is looking out to the players already owned by the club in order to make sure he has enough options available to him for the weeks ahead, and one thing the squad does seem to lack is natural width - with Jordon Ibe the only recognised out-and-out winger who seems to be regular in the matchday squad.
Some will suggest the Reds need to go into the transfer market this January in order to strengthen, but it is quote clear the Liverpool manager is determined to give the players owned by the club a chance to prove themselves at the highest level.