The speedy playmaker has lit up Euro 2008, taking his side of underdogs to the brink of a final appearance.
"We are interested in Arshavin. But it won't be easy to sign him because all the teams want him now," Barcelona vice president Ferran Soriano told El Mundo Deportivo paper.
"Our interest isn't sudden since we have been following him with discretion for a few months now," Soriano said.
"We don't want to lose him," Zenit official Konstantin Sarsaniya said. "We will discuss the offer with the player after Euro 2008."
Arshavin gave an interview to the BBC recently in which he spoke of his admiration for the English Premier League, saying in English, in between cursing in his native tongue, that he considered it the best in the world.
However, he has also confessed to "supporting Barcelona from childhood."
"Barcelona are my dream, I never thought I would have a genuine chance to play for the club," he told Russian media earlier this week.
Russia face Spain in the semifinals of Euro 2008 on Thursday evening. The victors will play Germany in the final on Sunday.