Arshavin found himself out of favour at the Emirates and secured a loan move back to his former club in February.
The 31-year-old scored three goals in nine starts for the Russian club, and now they want to discuss terms with the north London outfit over a permanent switch.
That is unlikely to happen until after Euro 2012, with Arshavin set to captain his country in their first game of the tournament against the Czech Republic tomorrow.
Zenit chief executive Maxim Mitrofanov also said the move will hinge on whether Arshavin wants to stick it out in the Premier League or secure a route back home.
"Everything depends on Arshavin. If Andrey is ready to stay in the team and go on the field as a player, we are ready to discuss it," he said.
"Arshavin is part of our club but, with all due respect to him, he will only stay in the event that he proves himself useful as an active player. Not as a man making up the numbers."
Arshavin joined Arsenal in 2009 for a club record £15 million but never fully managed to find his best form, only firing in fits and spurts before he was loaned out earlier this year.
Before departing for Russia, he managed just two goals in 27 appearances for the Gunners.
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