The centre-back, on loan from Tottenham Hotspur, was one of Ange Postecoglou's key players in the double-winning season.
His consistent performances have attracted attention from numerous Premier League clubs.
Celtic are keen to hold onto the US international as they prepare for Champions League group stage football as Scottish Premiership champions.
According to a report in the states, the league winners have been given a deadline of June 15 to complete the signing of the 24-year-old.
The English-born defender has been recalled by USA national team as he looks to add to his eight caps, and Everton, as well as Premier League newboys Bournemouth and Fulham, have been credited with an interest.
Carter-Vickers, who built a solid partnership with Carl Starfelt, could cost anything from £6million with a year left on his Spurs contract. He’s played just five times for the North London side with Celtic his seventh loan spell.
