Galatasaray officials were not immediately available to comment on the reports, carried by broadcaster CNN Turk and various news websites.
The 46-year-old, who led the club to UEFA Cup success in 2000 as a player, began his second spell as Galatasaray coach in October after a poor start to the season.
But the team are now languishing in 11th place in the Turkish championship, 27 points behind leaders Fenerbahce.
Hagi, who won 125 caps for Romania and featured in three World Cups, played for several clubs including Steaua Bucharest, Real Madrid and Barcelona as well as Galatasaray.
He had previously coached the Istanbul team in 2004/05 when he steered them to victory in the Turkish Cup.
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