The Dutch striker has netted an impressive 22 goals with the Gunners, being the top scorer for this Premier League season and drawing the attention of Los Blancos.
According to Spanish newspaper Marca, Madrid boss Jose Mourinho is keen on snatching up Van Persie, who has less than eighteen months left in his current contract with Arsenal.
Despite the uncertainty of the Portuguese manager's future at the Santiago Bernabeu, the club may still proceed with the move regardless.
La Liga rivals Barcelona are also said to be after the 28-year-old, with intentions to reunite him with former teammate Csec Fabregas.
Club vice-president Josep Maria Bartomeu denied any talks between the North London club or the player's agent, but refused to put down rumours of a possible transfer at the end of the season.
Van Persie, however, has hinted in January that he will still be wearing the Gunners' jersey for the next season after claiming on how his family has settled well in England.