The 22-year-old, who has been capped three times by his country and can play either as a winger or a No 10, is currently with the Istanbul outfit following a summer move from Steaua Bucharest in his homeland.
While Morutan has made more than ten appearances for Galatasaray, he is not a first-choice pick under veteran manager Fatih Terim and media in Turkey suggest that Rangers view him as short-term replacement should interest in their own Romanian Ianis Hagi become firmer.
A number of clubs, including Galatasaray, have been linked with a move for Hagi and Gers boss Giovanni van Bronckhorst is reportedly eyeing up a replacement.
Galatasaray are reluctant to let Morutan leave permanently given they spent £4million for his permanent transfer six months ago, but would be open to a loan move to continue his development.
Such a scenario, however, appears to hinge on concrete interest in Hagi, who has admirers across Europe.