Reports from Seoul claim that a fee of £7m has been agreed between the clubs and that only the finalising of personal terms is holding up the transfer.
Liverpool were the latest club to be linked with Ki, who has also attracted interest from the Bundesliga. Celtic received an offer from Russian club Rubin Kazan, believed to be £6m. The offer was rejected by the Parkhead side.
QPR made their interest known several weeks ago and, having now acquired the services of another South Korean, Park Ji-Sung, are confident they are very close to completing a move for Ki.
A source close to Ki is quoted as saying: "QPR has completed a deal with Celtic for Ki Sung-Yueng's transfer and the two sides are currently negotiating over his salary. Ki Sung-Yueng's side and the clubs are all positive about the transfer, so it is virtually a done deal. Ki will play in the English Premier League."
Neil Lennon will use the transfer fee to supplement his squad with new arrivals, with the Celtic manager tracking Ki's Olympic team-mate Kim Bo-Kyung.