Sky Sports News is reporting that the 22-year-old is keen to leave Liverpool before the European transfer window closes. He has been the subject of interest from teams in Belgium and France, while the Gers are still keen on bringing him back to Scotland.
Liverpool's preference is to sell Kent, who scored six goals and notched four assists during his loan spell with the Light Blues last season.
But the Reds are believed to value the winger at around £7 million, and aren't open to the idea of sending him on loan again.
Anfield boss Jurgen Klopp said earlier this summer: "[The loan] helped a lot. He came back a completely a different player.
“It was a very important year. Good club, good manager and good competition. Exactly the right thing for him at the right moment.
"But going back? [Only] if Rangers can buy him. The time is over giving him on loan.
“At a specific age, a player needs to find a place to settle and stuff like this. But we will see. A loan is not planned.”
Kent has been left out of Liverpool Under-23s last two matches amid reports he would "down tools" in a bid to force a move.