The former Chelsea striker moved to the Chinese side after his exit from Stamford Bridge this summer, but it has been speculated that his £200,000-a-week contract could be under threat due to a dispute between the club's shareholders.
Liverpool and Juventus were linked to the 34-year-old Ivorian in the closing stages of the transfer window, while the Sun claims Real were also monitoring events.
With Olivier Giroud and Lukas Podolski signed in the early part of summer before Van Persie was sold to Manchester United, Arsenal manager Arsene Wenger elected against further reinforcing his front line before the close of the transfer window.
Yet, with goal-scoring a problem in the early part of the season, Wenger could decide to act. If Drogba's contract was cancelled, he would be available outside the transfer window.
Podolski scored his and Arsenal's first goal of the season against Liverpool last time out, but Giroud has missed some notable chances early in his career at the Emirates.
The France international confirmed this week that he was struggling mentally following his move from Montpellier.
"I've been freezing up a little bit just when I've been about to pull trigger and that's exactly what happened in my first few matches for France," he said, according to Sky Sports.
However, he was confident that things would soon click into place.
"I'm not too worried about it, because I've been scoring in training and I feel pretty comfortable being a part of this team."
A move for Drogba would be unlikely for a number of reasons, however. Most notably, if he were to leave Shanghai then it is likely he would be looking for a salary comparable to what he makes in China-something Arsenal's wage structure wouldn't allow.
His abrasive, physical style that for so many seasons caused Arsenal all manners of problems in matches against Chelsea also would be a surprise fit in a Wenger side.
Perhaps a more likely option at some point down the line is Atletico Madrid's Adrian Lopez. The Spain international has indicated he is in no hurry to sign a new contract at the Vicente Calderon and, according to TalkSport, both Arsenal and Tottenham are ready to swoop.
The 24-year-old has an €18 million (£14.4m) buy-out clause in his current deal.
With Adrian also capable of playing on the right of the attack, he could potentially be seen as a replacement for Theo Walcott.
The England international has rejected Arsenal's offer of a new contract and his future at the Emirates looks precarious with just 12 months remaining on his current deal. While Wenger has claimed he believes Walcott will remain at the club, the winger would be free to negotiate a pre-contract agreement with a foreign club from January.
That would mean that the next transfer window would be the last opportunity to receive a fee for the 23-year-old. The Sun claims that Juventus could test Arsenal's resolve with a £10 million offer.
The story quotes an alleged Juventus inside, who says: "Juve have been interested in Walcott for some time but we are coming to the decisive moment."