The Scottish Sun says QPR, Rubin Kazan and Deportivo La Coruna are all keen to land the 23-year-old.
Rodgers is on the hunt for midfielders to suit the passing style he wants his side to play.
Ki's age and ability make him a leading contender to team up with Kop and England skipper Steven Gerrard.
QPR will not give up without a fight.
Chairman Tony Fernandes said last night: "Ki is a very good player and we are very aware of him.
"He is a player we will be watching and also monitoring closely."