The date coincides with the closing date for the announcement of candidacies for the IMF job, though Lagarde has not yet declared whether she will put herself forward.
The Court of Justice of the Republic, which is charged with deciding whether serving ministers can be probed, will meet on that date.
Although Lagarde has the support of France, Germany and Britain, her possible candidacy is dogged by accusations that she exceeded her authority by cutting short a legal battle between French tycoon Bernard Tapie, and a formerly state-owned bank, sending both parties into binding arbitration.
The arbitration panel decided to award Tapie, a supporter of Lagarde's boss President Nicolas Sarkozy, 385 million euros in the case, linked to the bank's alleged mishandling of the entrepreneur's sale of the Adidas sportswear firm.
The court could give one of three decisions on 10 June: it could throw out the case, it could ask for more information, or it could order an enquiry which in turn could see Lagarde charged with a criminal offence.
If convicted she could face five years in prison, though few expect such a scenario.
Dominique Strauss-Kahn resigned as IMF boss last week, saying he needed to fight charges including sexual assault and attempted rape of a hotel chambermaid in New York.