The Red Devils crashed out of the Champions League after a shock 2-1 defeat at home to La Liga club Sevilla on Tuesday night and have only the FA Cup left to play for this season.
De Gea, 27, essentially has two years left on his deal with United expected to trigger a 12-month extension clause in his current contract which runs until 2019.
The Spaniard signed his current contract in 2015 after a dream move to the Bernabeu broke down in the final minutes of the summer transfer window but Madrid have never ended their interest in the former Atletico Madrid stopper.
United held preliminary talks with De Gea over a new deal in December but Tuesday’s Champions League exit has left the Spaniard questioning his options, according to Goal.
Real Madrid have been keeping a close eye on De Gea’s situation at Old Trafford but are also interest in Chelsea stopper Thibaut Courtois.
The Belgian is keen on a move back to La Liga and said recently that his ‘heart is in Spain’.
Courtois’ wife and two children still live in Spain after his three years with Atletico Madrid.