The Russia international told reporters that Fabregas handed out end-of-season presents to his team-mates and refused to confirm or deny whether they were goodbye gifts when questioned.
"Only Cesc should decide if he leaves or not," Arshavin said. "Serious conversations on this topic in the team were not made - well, we joked a bit, as usual.
"Before leaving on vacation, Fabregas gave little gifts to each player and I asked him, 'What is this? A parting gift?' He said nothing. It says there is something behind this silence."
Despite Fabregas coming off of a disappointing season that saw him bag just three league goals, he is still waiting to see if European champions Barcelona will come calling for his services again.
The Catalans failed with two bids of £30 million and £35 million last summer and officials of the Catalan giants are due to discuss their summer transfer targets this week.