The 19-year-old striker, rated as one of the best teenagers, looks set to join the Spanish giants, who are reported to be willing to pay €60 million for the Brazilian's services
The striker has been heavily linked with a move to Europe for months, with the Madrid club and their rivals Barcelona believes to be leading the chase.
However, it appears Los Blancos could have won the chase for the highly-rated forward. The newspaper reports that the club's executive directors arrived in Brazil on Saturday alongside a representative from a Spanish bank as they look to agree a transfer fee.
The officials will meet with Santos president Luiz Alfaro Ribeiro de Oliveira on Sunday, in an attempt to close a deal.
Madrid would pay €60 million in total for the 19-year-old with €20m paid upfront.
Neymar has made 42 appearances for Santos this season, scoring 21 goals in the process.