The Brazil international has been tipped to exit the Parc des Princes following just one campaign after supposedly growing unhappy with life in Paris.
Madrid have been monitoring the situation with the intention of swooping in the future, though they were expected to delay an official approach until at least 2019.
Neymar, who ended his four-year stay at Barcelona to join PSG for a world-record sum in August, is said to favour a move to the Bernabeu over any other possible destinations.
According to Goal.com, PSG president Nasser Al-Khelaifi is happy to sanction the move as early as this summer if Neymar fulfils his promise of bringing the Champions League to the French capital for the first time.
Having spent three weeks on trial with Los Blancos in 2006, the 25-year-old is said to have unfinished business and is desperate to swap clubs at the earliest possible chance.