The attacking midfielder is set to leave Parc De Princes this summer, as the deal is now at the 'final stages', as per transfer expert Fabrizio Romano.
The deal between the two sides is worth a reported 35 million euros.
Atléti are keen to get the deal over the line quickly as Diego Simeone looks to bolster his squad ahead of the new La Liga season and for their Champions League campaign.
The 25-year-old looks set to become the Madrid club's second signing of the summer transfer window, having already brought in left-back Alex Grimaldo on a permanent deal from Bayer Leverkusen this summer.
The South Korea international made three appearances at this summer's World Cup and provided one assist as his nation were knocked out following the group stage.
But the two time Champions League winner is set to put that disappointment behind him as he gears up to play his football at the Metropolitano stadium next season.