The channel did not give any more information and Galatasaray could not be reached for immediate comment.
The 34-year-old, who is on Africa Cup of Nations duty with his country, moved to China after helping Chelsea win the Champions League last May.
Drogba signed a two-and-a-half-year deal with Shenhua in June that was, according to media reports, worth around $300,000 a week.
However, the Shanghai-based Oriental Sports daily said last month that the player was owed money by the club.
Drogba applied for permission to leave on loan before the January transfer window but world football's ruling body FIFA refused the request.
The striker returned to Chelsea to train at his former club in a bid to stay sharp for the Nations Cup, fuelling media speculation of a return to Europe.
If his move to Turkey is confirmed, the transfer will come less than a week after Netherlands midfielder Wesley Sneijder joined Galatasaray from Inter Milan.
Galatasaray have been drawn to meet German club Schalke 04 in the last 16 of the Champions League next month.
Drogba's former Chelsea team-mate Nicolas Anelka ended a one-year spell at Shenhua when he joined Italians Juventus on a five-month loan deal on Saturday.
The signings of Drogba and former France striker Anelka were heralded as a coup for the Chinese Super League last year.
After years of negative headlines brought by match-fixing scandals which hit the league's credibility and led to jail terms for former Chinese football chiefs, the two transfers drew attention for all the right reasons.
Drogba scored 157 times in 341 appearances for Chelsea, making him the club's fourth all-time leading scorer.
Having left Olympique Marseille to join the Londoners in 2004, he won everything in England and his strength and speed made him a handful for world-class defenders.
He was Chelsea's Champions League talisman last season, scoring a brilliant headed equaliser in the final against Bayern Munich to force extra time and then scoring the winning penalty in the shootout.
