The Daily Mirror reported that Arshavin's 19 million-pound ($38 million) move to the North London club was "being finalized" subject to the 27-year-old being granted a work permit.
Zenit had earlier turned down a 15 million euro ($23.4 million) offer from Barcelona as being "too low."
The Russian Sport Express newspaper quoted Arshavin as saying in response to a question about his future that he had "made a decision," before adding "I can't say anything right now."
Tottenham, who sold Irish international Robbie Keane to Liverpool on Monday for some 20 million pounds, have also been linked with Spartak Moscow striker Roman Pavlyuchenko. The current Spartak captain has apparently been lined up as a possible replacement by Tottenham for the Bulgarian Dimitar Berbatov, rumored to be on his way to either Manchester United or Barcelona.
Both Arshavin and Pavlyuchenko starred as Russia unexpectedly made the semifinals of Euro 2008 last month.