United look to have won the race for the France Under-21 star, who has been also been linked with Inter Milan, AC Milan and Arsenal.
Reports in France say a £3million fee has been agreed and Obertan will sign a four-year contract.
Bordeaux sent him on loan for the second half of last season to Lorient where he made 15 appearances, scoring one goal.
Club president Jean-Louis Triaud stated yesterday it was a "great possibility" United would sign Obertan.
Obertan scored against England Under-21s in March and took the Most Valuable Player award at last month's Toulon International youth tournament.