The 57-year-old coach won the Ligue 1 title with Bordeaux in 1999 but he was sacked by Nantes following their relegation from the top flight three years ago
The 57-year-old coach, who has been out of the game since being fired by Nantes three years ago, has put pen to paper on a contract which ties him to the Stade Velodrome until the summer of 2014.
"Elie Baup is named coach of l'OM for two seasons," a statement on the club's official website read.
"He should now travel to Crans-Montana, where the players have been in training since Wednesday. He will be presented to the press on Thursday afternoon."
Baup won the Ligue 1 title with Bordeaux in 1999 and drew plenty of plaudits for guiding Toulouse into the qualifying stages of the Champions League on the back of a stunning third-place finish in the top-flight five years ago.
However, the club only just beat the drop the following season and he took Nantes down after moving to the Beaujoire-Louis at the start of the 2008-09 season.