Dzagoev is in the final year of his contract at CSKA Moscow, but isn't known to crave a move away, despite the reported interest from Arsenal.
In both of Russia's opening fixtures, Dzagoev has played an attacking midfield role, and is already one of the players of the tournament.
Arsene Wenger has already brought in Lukas Podoslski to strengthen The Gunners' attack, while a move for Montpellier striker Olivier Giroud is understood to be in the offing.
Rennes midfielder Yann M'Vila has already been mentioned as a potential recruit, to assist in bolstering Arsenal's midfield, which has seen Jack Wilshere miss the whole of last season.
The CSKA man would join Andrey Arshavin in north London should he move to the Premier League; however his Russian team-mate could make a move back to Zenit St Petersburg on a permanent basis after last season's temporary loan spell.
CSKA are understood to value Dzagoev at £13m, however with his contract due to expire at the end of the year, would be expected to accept a reduced offer.
Regularly frugal in previous transfer windows, Arsenal are likely to be attracted by a reduced fee for a player who is likely to be hot property after Euro 2012.
However, Wenger will be wary of buying a player who has stared on a short-term basis, and note that while the performances of Arshavin have also been impressive this summer, his club showings have faied to live up to his reputation.
Arshavin has flattered to decisive since joining from Zenit in 2009, and could have played his final game for Arsenal.
As well as signing new players, the club's main priority this season surrounds the immediate future of Robin van Persie, who is yet to sign an extension to his contract following a 30-goal season for the Dutchman.
To contact the editor, e-mail:
Most Popular Slideshows
- 'Game of Thrones'-like Film, 'The Queen of the Tearling,' Casts Emma Watson as Lead Star and Exec Producer [PHOTOS]
- Kim Kardashian Baby Girl: Suggested Ways Kanye West’s New Born Can Earn Money to Keep Up with the Kardashians [PHOTOS]
- Asus Transformer Infinity Pad, Sony Vaio Duo, Toshiba Satellite, A Look at Intel's Haswell 4th Generation Ultrabooks and Notebooks [Photos]
- 'Fifty Shades of Grey' Movie Casting: Selena Gomez Open to Nude Role, Ready for Anastasia Steele? [PHOTOS]