The Premier League duo were reportedly interested in the 19-year-old left-back, and were believed to be scouting the youngster, but Real Madrid's defensive troubles are thought to have prompted the La Liga giants to enter the race for Digne.
The Lille defender has established himself as first-choice in the Ligue 1 side this season, and his impressive performances in the Champions League have attracted interest.
Chelsea left-back Ashley Cole is out of contract at the end of this season, and the England international has been linked with a move away from Stamford Bridge - Real Madrid and Paris St Germain have both been credited with an interest.
Cole's deputy, Ryan Bertrand, recently signed a five-year deal at Chelsea, and is highly-regarded at Stamford Bridge, but if Cole departs, Roberto Di Matteo would likely look for additional cover.
Manchester United have also looked at Digne, as they search for a long-term replacement for France international Patrice Evra.
Sir Alex Ferguson signed Dutch fullback Alexander Buttner in the summer, but the opporuntity to sign Digne, one of the premier young talents in his position in the world, could be to good to miss for the United boss.
But Real Madrid, after watching Marcelo and Fabio Coentrao pick up injuries, are now thought to be considering a January swoop for the French teen, with a view to beating Manchester United and Chelsea to the punch.
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