According to the News of the World, Wenger has been a long-term target of Real, with president-elect Florentino Perez desperate to lure the Frenchman to the Bernabeu.
Suggestions are that Wenger would have been offered a five-year deal, with bonuses for winning the Champions League and Primera Liga. In total, this would have added up to almost £8m a year.
However, the NotW claims Wenger has decided to stay at the Emirates and sign a new contract that will tie him down for at least another three years.
Arsenal's board are desperate to agree a deal with Wenger as soon as possible, knowing full well the interest from Real and Bayern Munich.
There are now major funds at Wenger's disposal at Arsenal with Stan Kroenke, who has recently become the club's major shareholder, making a £40m transfer kitty available. This would be inflated even further should Emmanuel Adebayor be allowed to join AC Milan this summer.
Arsenal head into their all-or-nothing Champions League clash with Manchester United on Tuesday secure in the knowledge of another top-four Premier League finish after their 3-0 victory over Portsmouth on Saturday.
Victory meant Arsenal have now gone 21 matches unbeaten in the league, despite fielding an under-strength team against Pompey, and Wenger confirmed his team selection was nothing to do with any fresh injuries.
"I wanted to give a breather to some of the players," he said. "We dominated the first half and created plenty of chances through our movement and technical quality.
"We were sharp, composed and mobile with a great attitude. They changed their game plan at the start of the second half and we struggled a bit.
They had the turning point with a potential 2-1, but after we scored the third goal, the game was over.
"I have no doubts about the younger players because I see them every day and what we work on, you could see that on the pitch. Some of those who played could play again on Tuesday."
Portsmouth were less than impressed by the decision to award a penalty when Andrei Arshavin went down under a challenge from Sean Davis.
Wenger, though, maintained the decision by referee Lee Mason was correct.
"Some people said Andrei said it was not a penalty, but I do not know," he said. "From outside, it looked a penalty, but before there was 100 per cent a penalty on (Carlos) Vela."
Wenger revealed the decision to hand Arshavin the captaincy against Portsmouth was reward for the way the Russian has settled into the squad following his January deadline-day move from Zenit St Petersburg.
"I like rotational leadership and he was the most experienced player in a young side," said the Arsenal manager.
"Since he has arrived here, he has a fantastic attitude with everybody and I wanted to reward that."