The 22-year has recently broken into the Brazilian outfit's first-team plans, but Carlo Ancelotti's side have tabled a bid which would involve Santos initially joining on loan to with a view to a permanent deal.
Santos would primarily be given the chance to impress playing Segunda football in the club's Castilla team, and he revealed that Real scouts had been monitoring his progress.
"I received the offer before the game against Barcelona, they watched me in three games in a row when I played as a starter against Portuguesa, Coritiba and Crac," Santos told Globo Esporte.
"I am very happy, I am a reserve player and Real Madrid are interested in me.
"It's of a course a dream, but it's in God's hands. If I have to go now, I will. If not, I can go later."