Poyet hopeful of long-term Ki deal
The 24-year-old signed a season-long loan at Sunderland in August, but Swansea City had the option to recall him mid-campaign if he impressed.
Swansea decided not to ask him to return to the Welsh club, though, despite the South Korea international playing a big role in Sunderland's revival under Poyet.
The Stadium of Light outfit have climbed off the bottom of the Premier League table and lead their League Cup semi-final against Manchester United 2-1 after the first leg.
"There is a chance (of a full transfer)," Poyet said.
"Swansea had an opportunity to call him back, and they didn't. That was a difficult day for me because I was waiting and waiting.
"I think there are a couple of things we need to make sure: where we are and where we want to go, and with the player as well.
"I can see that he's enjoying his football and making a difference."
Ki scored the third goal of his loan stint on Saturday as Sunderland won 4-1 at fellow strugglers Fulham.
Poyet was delighted that he netted again and hailed Ki as his best player since taking over from Paolo Di Canio in October.
"He's very happy, it means so much for him to score that goal," he added.
"He cares and he's enjoying his new position a bit further up the pitch and making a difference. I want players with that mentality, to make us play, and he's been excellent.
"He's been the most consistent player for me, in my way of playing."