The Gunners attacking midfielder, who can also play on the wings, has attracted the attention of the European giants following a number of impressive performances for Arsene Wenger's side this season.
And Arsenal are at a risk of losing him after failing to tie him down to a long-term deal. Nelson's contract expires in the summer of 2019 and the club are yet to offer him a new deal which has alerted interested suitors.
Real and PSG are monitoring the situation and could make a move in the summer, when he will have 12 months to run on his current deal. The 18-year-old has made his debut in the Community Shield at the start of the season and he has followed it up with 10 more appearances with the first-team.
Wenger has mainly used the wide attacking player as a right wing-back, and despite playing out of position, he has impressed every time he is given an opportunity. He has been mainly used in the Europa League and the Carabao Cup, making an appearance in every game except the semi-final first leg against Chelsea in the latter.
Nelson is highly rated by the coaches at London Colney and according to the Daily Mail, there has been no progress made in terms of tying him down to a new long-term deal. Interest from the likes of Real and PSG is likely to spark the north London club into action, if not they could risk losing him either in the summer or next year on a free transfer.
Arsenal are no strangers to the situation as they have a number of key first-team players in the final 18 months of their contract, namely Aaron Ramsey, Olivier Giroud and Nacho Monreal, while the likes of Alexis Sanchez, Mesut Ozil and Jack Wilshere have just over five months remaining on their deals.
The Chilean international looks certain to leave this month with Manchester United leading the chase, while Ozil is expected to leave on a free transfer in the summer. Wilshere is the only one among the trio who confirmed his interest in committing his future to Arsenal and the club are said to have offered him a new deal.