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 Defence Minister Jean-Yves Le Drian was expected to visit the scene early Tuesday.
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).
Ukrainian man, French teen killed in French Alps avalanche
French prosecutors have opened an investigation into the events to see whether the probe should be handed over to military authorities.
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.
Monday's accident came after two French teenage students and a Ukrainian tourist were killed in an avalanche in the French Alps last Wednesday.
A French teacher who took the students onto a closed skiing piste in that incident has been charged with involuntary manslaughter.
The 47-year-old, who was seriously injured in the incident was charged at his bedside in hospital in the south-central city of Grenoble.
The victims, a 16-year-old girl and 14-year-old boy, were from a school in Lyon. The 56-year-old Ukrainian man who was also killed in the avalanche was not part of the group.
According to prosecutors, the piste was closed off with netting and warnings in four languages but the school group climbed over the netting "fully aware" of what they were doing.
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