Zenit St. Petersburg rejected a 15 million euro ($23.4 million) bid for Arshavin on Monday, stating on their website that the offer was "too low".
"Of course, I'm aware of what has happened," Arshavin told the Russian Sport-Express paper. "But, hey, no is no. I fully understand the position of Zenit. The sum offered was, I suppose, really too low. It means my dream will remain a dream. Or come true at another time."
Arshavin has spoken more than once of his love for the Spanish side, and has supported them since childhood.
"I want to leave Zenit," he went on. "As for offers from Chelsea and Arsenal, I'm not in a position to divulge that information right now."
The architect of Russia's victories over Sweden and Holland at Euro 2008, Arshavin was philosophical about the role his performances at the tournament had played.
"I have the desire to play in a stronger championship. It's worked out strange though - my good performances at Euro 2008 have meant that my price has gone up," he said.
Arshavin scored two goals in three games at Euro 2008, and was named in UEFA's symbolic squad for the tournament along with fellow Russians Roman Pavlyuchenko, Yuri Zhirkov, and Konstantin Zyrianov. However, one of the factors behind Barcelona's reluctance to up their bid for the 27-year-old Arshavin is believed to have been his ineffectiveness against Spanish La Liga defenders as Russia went down 3-0 to Spain in the Euro 2008 semifinals.
Arshavin missed Zenit's 5-1 victory over Tom Tomsk on Sunday, with the side's Dutch manger, Dick Advocaat, saying that he would not play for the team until the situation was "cleared up."