The ministry did not clarify where the five people have been travelling.
The mosquito-borne virus has been linked to serious birth defects, including microcephaly, a condition in which a newborn's brain and skull are abnormally small. Developmental problems often result.
The Zika virus is transmitted by the Aedes aegypti mosquito. The insect can also carry dengue, chikungunya and yellow fever.
"There is no actual risk of transmission of the Zika virus in mainland (France)," the ministry added. "We are not in a period of activity for the Aedes mosquito, the vector of the virus, which falls between May and November."
The World Health Organization (WHO) set an emergency meeting for February 1 on the Zika virus, which it says is spreading "explosively" in the Americas, with three to four million cases expected this year.
France earlier on Thursday urged pregnant women not to travel to French overseas territories in South America and the Caribbean, including Martinique and French Guiana where epidemic have been declared.
Guadeloupe and Saint Martin have also each reported one confirmed case.
The health ministry has said that mainland France would be sending medical staff to Martinique and Guyana in the coming days to evaluate the needs of hospitals and doctors.