The 23-year-old had been all but set to sign for the Parkhead side with numerous reports suggesting the deal had been done.
English Championship side Middlesbrough have made a late bid to sign McGree.
Boro are reported to have made a more attractive offer to both player and club.
The six-time Australia international is well known in the Championship having spent time on loan at Birmingham City, making 30 appearances for the Blues.
Celtic boss Ange Postecoglou is an admirer of McGree having given him his first call-up for the Socceroos and was hopeful of getting a deal done for the midfielder who can play different roles in the centre of the pitch.
Middlesbrough have also targeted Rangers centre-midfielder John Lundstram.
The 27-year-old has been a bit-part player since Giovanni van Bronckhorst took over and Boro boss Chris Wilder is keen to link up with him once more.
Wilder was his boss at Sheffield United where Lundstram played 120 times and helped the Blades win promotion from the Championship.
Meanwhile, Dundee are keen on Celtic defender Osaze Urhoghide.
The Dens Park side have the worst defensive record in the Premiership with James McPake keen to strengthen in that area of the pitch before the transfer window closes.