Another six soldiers were injured near the resort of Valfrejus, with one in intensive care in hospital after suffering hypothermia.
They were among a group of 50 soldiers taking part in the skiing exercise.
Three helicopters were involved in the search, as well as five police dogs and three doctors.
The nationalities of the victims were not immediately known. The French Foreign Legion attracts recruits from around the world.
French President Francois Hollande expressed "the nation's solidarity" over the deaths and wished the injured a speedy recovery.
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The avalanche took place at 1:50 pm local time (1250 GMT) at an altitude of between 2,350 and 2,600 metres (7,700-8,500 feet).
French prosecutors are looking into whether they or military authorities should handle the probe of the accident. If investigators find evidence of criminal wrongdoing the military would handle the case.
The regiment to which the soldiers were attached is specialised in mountain warfare and all the members have military skiing qualifications.
The unit experienced a similar tragedy in 2012 when an avalanche swept away five of its members, leaving one dead.