The 31-year-old is surplus to requirements at the Emirates and spent the second half of last season on loan at Zenit St. Petersburg, from whom he joined the Gunners in 2009.
However, his former employees are not believed to be keen on offering Arshavin a permanent return to Russia. Nor are Anzhi Makhachkala, who have been credited with an interest in the past.
Dynamo Moscow appears to be the most likely destination for the forward, but neither they nor rivals CSKA have yet made an offer.
Arsenal have included Arshavin in their 25-man Champions League squad but are considering giving him a free transfer in order to get him off the wage bill, according to The Telegraph.