Evra, 37, is a legend at Old Trafford having spent eight years at the club as a player.
During that time he won five Premier League titles, three League Cups and the Champions League.
Evra, who spent the end of last season at West Ham, is now without a club and he has been pictured alongside Woodward at numerous games this season.
The Mail claim Manchester United could now hand him a permament job in an ambassadorial or liaison role.
Evra has been allowed into the dressing room by Jose Mourinho and he is said to get on well with many of the first team players, including Paul Pogba and Anthony Martial.
It could even be a way of helping the pair’s relationship with the under-fire boss.
United have a track record of bringing in their former players into the club’s off-field activity once they have hung up their boots.
Gary Pallister, Paul Parker and Dennis Irwin have all been employed by United once their playing days are over.
Patrice Evra – the Manchester United legend’s honours list
Premier League – 5
2006–07, 2007–08, 2008–09, 2010–11, 2012–13
Champions League – 1
League Cup – 3
2005–06, 2008–09, 2009–10
Community Shield – 5
2007, 2008, 2010, 2011, 2013
FIFA Club World Cup – 1