Dzagoev has enjoyed a blistering start to the European Championship and has scored three goals in Russia's opening two games against the Czech Republic and co-hosts Poland.
The 21-year-old has long been touted as one of the best prospects in European football and he is out of contract at CSKA in December.
Arsenal have most recently been linked with a bid for the attacker, although Dzagoev has previously suggested he would be interested in joining Chelsea if he were to move to the Premier League.
He remains keen to eventually play for a different club, but is happy to put thoughts of a transfer on the back-burner until Russia's involvement in Euro 2012 is over.
"Of course, I have the desire to try myself in other clubs in Europe," said Dzagoev prior to Russia's final group game against Greece on Saturday.
"But I'll think about it only after the tournament. Now I'm doing my own job here."
Dzagoev also believes that, despite scoring three times and earning plenty of plaudits, he has yet to hit his best form in the tournament.
He added: "I don't think that I've played two good games. Yes, I scored in the second game, but it wasn't as good as the first game. I lost a lot of balls."