AS said that Real would initially seek to bring Arshavin to Madrid on loan until June 2009, and then buy him outright for around $40 million.
Real have looked short of attacking power of late, and the club are reportedly looking to add a creative forward to their line-up.
Arshavin, 27, became a target for a number of European sides this summer after helping Zenit lift the UEFA Cup and then propelling Russia to the semifinals of Euro 2008.
Although the diminutive forward made no secret of his desire to "test himself abroad," Zenit, controlled by state-run energy giant Gazprom, subsequently turned down offers of some $20 million and $26 million from Barcelona and Tottenham Hotspur respectively as "too low."
However, the player told the Sport Express paper last week that, "Four months ago I told the president of the club that I didn't want to remain in the side any longer. I then repeated this more than once. Unfortunately, my words were either not heard, or not taken seriously."
"This is all heading to one thing - for Zenit to name a reasonable price. If they do this, then, of course, I'll leave. If they don't, I'll be listed as a Zenit player, but on paper only. I won't play here next year," he said.
AS reported that Madrid's sporting department informed club president Ramon Calderon of Arshavin's availability after reading the interview.
Arshavin, who came through the club's youth system to become a genuine European star, has spent his entire career at Zenit.
Zenit trainer Dick Advocaat has also backed Arshavin, saying that, "It is necessary to allow Arshavin the opportunity to move on."
"We have had some good moments together but it is now time to go our separate ways. It is best for both the player and the club if he can sign for one of Europe's great teams," added the Dutch trainer.
Tottenham Hotspur are also expected to renew their interest in Arshavin.
AS also said that Zenit club representative Alexei Petrov has told clubs they should start preparing bids for the 2006 Russian player of the year.
"In January, we will not negotiate through any of Arshavin's agents. Instead, we will deal directly with the clubs who are interested in him," said Petrov.
If Real Madrid were to sign Arshavin, the Russian would be unable to play in any of his new club's Champions League games as he is cup-tied after playing four matches for Zenit in the competition.
Zenit are in the same group as Real in Europe's most prestigious club competition. The two clubs clashed on September 30 in St. Petersburg, Zenit losing 2-1 despite dominating their illustrious opponents for much of the game.
The return fixture is on December 10 in Madrid, and could decide which of the two clubs joins group leaders Juventus in the play-off stages of the Champions League. It could prove an opportunity to clear up the long-running Arshavin transfer saga.