Drogba agreed to an 18-month contract with the Istanbul side on Monday, in a deal which includes a 6 million euro transfer fee and a 4 million euro signing on fee.
However, Shanghai Shenhua said the agreement had violated the club's contact with Drogba and they were ready to raise the issue with world governing body FIFA.
"The club is deeply shocked," the Chinese Super League team said in a statement on its website.
"Drogba is still a Shanghai Shenhua Football Club member as the contract between the two parties is still within its period of validity," it added of the deal which saw Drogba sign a two-and-a-half-year contract last July.
"The Shanghai Shenhua Football Club is ready to gather evidence and submit the findings to FIFA in order to protect its interests."
The 34-year-old Ivorian, currently on African Cup of Nations duty in South Africa, opted to play in China after leaving Chelsea following their Champions League triumph last May.
Before signing for Galatasaray, Drogba had applied for permission to go out on loan during the Chinese off-season but FIFA refused his request.
Last week, fellow Shenhua striker Nicolas Anelka joined Italian giants Juventus on a short-term loan deal.