In a statement, Marseille said both clubs had agreed to call off the clash at the Stade Velodrome, which was scheduled to take place on Wednesday, "as a health precaution".
One unidentified Montpellier player has tested positive for COVID-19 and will be tested again on Friday.
If the case is confirmed, it will be the fifth player in Michel Der Zakarian's squad to be affected by the disease since the start of the pandemic.
Midfielder Junior Samba, who was placed in an induced coma at the end of April, was the first Ligue 1 player to test positive for the virus.
The cancellation comes following Montpellier's friendly last week with Strasbourg, who a few days after the match announced five of their players had tested positive for the virus.
Andre Villas-Boas's side are set to open the new term by hosting Saint-Etienne on August 21.
Later on Tuesday Strasbourg announced their pre-season fixture against Reims on Saturday has been cancelled after the Alsace side recorded four new cases of the illness within their squad.
Last week they said five players had tested positive which led to the calling off of last weekend's match against Nancy and Tuesday's game against Dijon.
"Unfortunately these tests confirm our worries, but we worked in tight collaboration and in total transparence with the regional health board," Strasbourg's doctor Francois Pietra said in a statement.
Strasbourg begin the new campaign on August 23 by travelling to newly-promoted Lorient.