The 24-year-old is currently on a season-long loan from Tottenham Hotspur and has emerged as a central figure under Ange Postecoglou.
He has played 36 times for the club in six different competitions.
In December the Australian confirmed he was "absolutely” keen to keep the US international beyond this season.
“I wouldn’t have brought him in if I thought it was just a short-term thing,” Postecoglou said.
“He has become an important part of our team and that’s why I was so keen to bring him in.
"The beauty of that is he is only going to get better. And he wants that. He works hard every day, looks after himself and wants to be the best footballer he can be. I knew what I was getting but it has turned out to be a really good move.”
It has been reported that it could cost as much as £6million to make the deal permanent in the summer with no clause in the deal.
According to reports in the US, four English sides have joined the race for Carter-Vickers’ signature following his impressive campaign at Celtic Park.
Leicester City, managed by Brendan Rodgers, and Wolverhampton Wanderers are the two main interested parties.
Carter-Vickers, who has eight international caps, is on his seventh loan spell away from Spurs having had stints at the likes of Sheffield United, Swansea City, Stoke City and Bournemouth.