The southern French club suffered their fifth straight defeat in all competitions on Saturday with Reims claiming a shock 3-2 win at Marseille's Stade Velodrome.
Marseille came from two goals down at half-time to equalise with four minutes to play but a stoppage-time header from Prince Oniangue handed the visitors victory and increased the pressure on Baup.
Despite the result, Marseille's sporting director Jose Anigo argued the club remain confident Baup can turn the nine-time French league champions' form around.
"(Baup) is supported by everyone and we have a constant dialogue," Anigo said.
"He must be able to work calmly."
Marseille's president Labrune Vincent was similarly supportive of Baup.
"We are together and together we will find solutions to get out of this," Vincent told L'Equipe.
Saturday's result saw Marseille slip to sixth in Ligue 1, seven points behind leaders Paris Saint-Germain, who have a game in hand.
Marseille have not won since September 27 when they triumphed 2-0 away to Lorient and have since lost twice in the UEFA Champions League and three times in Ligue 1.
While the French media are starting to ask questions of Baup, who has been head coach at the Stade Velodrome since July 2012, Marseille captain Steve Mandanda reasoned everyone at the club needs to take responsibility for their poor run of form.
"We all realise we are in a difficult situation," Mandanda said.
"We all need to be more committed and stand together but I believe in this team."
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