The Automobile Club de l'Ouest said Simonsen, driving the No.95 Aston Martin Vantage GTE, "exited the track at high speed" on the Tertre Rouge corner of the Le Mans track on the fourth lap of the GTE Am category race.
The club said Simonsen, 34, was immediately attended to by doctors on the scene and was quickly removed to the Circuit Medical Center, where he "died soon after due to his injuries."
The statement said David Richards, principal of the Aston Martin racing team, informed Simonsen's family of the death.
"The Automobile Club de l'Ouest wishes to express its great sadness following this incident, and extends its deepest condolences to the family and those close to Allan Simonsen," the officials said.
The racing body said it would make no further statement while the incident was being investigated.
The fatal accident came only two days after Simonsen and Danish teammates Christoffer Nygaard and Kristian Poulsen had claimed the pole position for their No. 95 Aston team, The Telegraph reported.