Gala signed Rodrigues in a 3.7million euros deal from Greek outfit PAOK last January and could have made a significant profit over the summer after Newcastle submitted a £6million bid, according to Turkish reports.
However, the Istanbul outfit opted to snub Newcastle's interest as they see Rodrigues as a crucial player in their bid to win the Super Lig this season.
But the Premier League side are planning an improved bid in January regardless and have sent scouts to keep tabs on the winger this season.
The 26-year-old Netherlands-born Cape Verde international is under contract with Gala until 2021 so they are under no pressure to sell, but Newcastle are hopeful a deal can be done.
It is understood the St James' Park side would be prepared to wait until the summer transfer window if Galatasaray cannot be convinced to part with Rodrigues in a couple of months.