The French starlet, 20, joined Marseille only last January from Lille despite interest from Arsene Wenger, who was keen to bring him to Arsenal.
Thauvin has continued to catch the eye at the Stade Velodrome and according to reports in France representatives of both United and Chelsea were in the crowd to watch him in action during Marseille's 2-1 home defeat to Borussia Dortmund in the Champions League on Wednesday.
The Ligue 1 club, amid building interest from around Europe in their young star during the summer, moved quickly to secure him on a long-term contract with Thauvin signing a new five-year deal in September.
However news of Chelsea and United's scouting mission could turn the French Under-21 international's head with the transfer window set to open.
Jose Mourinho and David Moyes are both looking to bolster to their squads with fresh attacking talent ahead of the final five months of the campaign.
Thauvin, who has also attracted the interest of Aston Villa, Manchester City and Newcastle, has made 12 league appearances for Marseille this season scoring twice.
