The 20-year-old patient was taken to Grenoble University Hospital from the city's main train station in the very early hours of Monday morning after complaining of dizziness and a high fever.
At 2:30am on Monday morning police set up a security perimeter around the station.
The patient, who has applied for refugee status in France, had been to the Congo and Sierra Leone, both areas blighted by the deadly virus, according to Le Dauphiné Libéré.
In a statement, the hospital said the patient had been admitted and transferred to Lyon's Croix-Rousse hospital “in accordance with national and local regulations”.
The hospital added that a previous patient with suspected Ebola had been admitted last Thursday and “isolated in a place specially equipped for this type of care”.
The epidemic, which has hit Guinea, Liberia and Sierra Leone hardest, has so far claimed around 11,178 lives, according to the Centres for Disease Control and Prevention.
There have been a total of 27,311 cases of the disease worldwide.
In Sierra Leone alone, there have been a total of 12,932 suspected and confirmed cases and 3,917 deaths.
If the case is confirmed this won't be the first Ebola patient to be treated in France. In November last year a United Nations employee who contracted Ebola in Sierra Leone was repatriated in France and treated at the Begin military hospital outside Paris.
A French nurse, who worked for Doctors Without Borders (MSF) in Liberia, was treated for Ebola at the same hospital in September and also recovered.