360 Sports reports that officials from the Turkish club are in London to negotiate a January transfer for the duo.
Galatasary currently are top of the Süper Lig on 40 points, four points better than last year's champions Fenerbahçe after 18 league games.
Despite their recent run of good form, which has seen them go on an eight match unbeaten run, they'd be looking to consolidate their superior position to ensure winning the title and playing in the Championships League next season.
Arshavin—a versatile player who can play in the middle as an attacking midfielder, or as a supporting striker as well as out on the flank, could help them achieve these aims.
For Arshavin, a new environment and the thrill of a new challenge could be the tonic he needs to revive his form, which of late has settled into a lethargic doldrums.
Arshavin recently admitted that his current form leaves a lot to be desired. Responding to criticism in late October, Arshavin said he understood fans' frustration.
"They should expect more from me, of course. They should expect some more tricks, goals and unbelievable passes.
"To be honest I am still expecting more from me as well. But believe me, I do everything I can do.
"I just have to give everything I can at the moment, just on the pitch. I want to score more goals and bring my team to the title."
Despite this admission, things are yet to turn around for him. A move to a different environment could help revive his sliding form.
Nicknamed "The Little Mozart" for his vision and ability to orchestrate play in the midfield, Rosicky came to Arsenal amidst high expectations, the fulfillment of which have been hampered by injuries at Arsenal.
He competes currently with Yossi Benayoun (on loan from Chelsea) to play second fiddle to Aaron Ramsey for the creative midfield position.
A move to Galatasaray could enable him to play consistently. Moreover, it just may be the right thing his career—already at the dawn of its twilight—needs.
Please read my Theo Walcott article here.