Brendan Rodgers is reportedly keen on the 21-year-old, who began his career in Rangers' academy.
And to win the race, Rodgers' Celtic will have to see off interest from a former Rangers boss, Nottingham Forest's Mark Warburton.
The Championship club reportedly are weighing up a £500,000 offer for the Scotland under-20 international.
Rangers too have been loosely linked, but it is Warburton's Forest who pose the biggest threat to Celtic's hopes of landing the talented midfielder.
With Morgan's Gers history, he could see a move to Forest as more hassle-free than joining Celtic.
Both Warburton and Rodgers have good reputations in terms of developing young players.
Signing him may not be easy, as Celtic have European football on offer and increased funds as a result.
If Warburton is successful it could be seen as a belated revenge over Rodgers, who wiped the floor with his Rangers side during their spell in opposition in Scotland, and hastened his Ibrox demise.