Celtic had looked set to snap up Australia international midfielder McGree from MLS side Charlotte FC, with an initial fee of just under £3m agreed.
McGree is now in the UK, but he has not yet completed a move to Celtic and the Scottish side are facing fierce competition from Middlesbrough.
Chris Wilder wants McGree in the north east and Middlesbrough are pushing hard for his signature.
It is suggested that Wednesday will move the transfer saga along, though whether it will be to its conclusion is unclear.
Wilder has Boro sitting seventh in the Championship table and shaping up for a shot at promotion to the Premier League this term.
The project at the club could appeal to McGree, but a move to Celtic would see him link up with countryman Ange Postecoglou.
McGree is familiar with Championship football due to having had an 18-month loan spell at Birmingham City.