The Brazilian international joined Spurs over the summer from Corinthians, and has made a good start to his career in London.
Despite only being a recent arrival at White Hart Lane, Real are already considering a move for his services in January.
The Spaniards were interested in the 25-year-old prior to his move to Tottenham, but failed to make a serious offer for him.
Los Blancos chief Carlo Ancelotti sees him as an ideal addition to his squad, and is hopeful of exploiting the partnership between the two clubs.
Spurs and Real are known to have close ties, which has contributed to deals for Luka Modric and Gareth Bale in recent years.
Andre Villas-Boas is desperate to keep Paulinho, and is ready to offer him a significant pay rise to stay at the club.
The midfielder’s current deal expires in 2017, but Tottenham are hopeful of adding another 12 months to that figure.
Paulinho cost £17m over the summer, and Spurs would likely demand a huge profit to sell him after only six months.
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