Israel said "initial indication" suggested that the solider had been seized while taking part in an operation to decommission a tunnel belonging to Hamas, Israeli media reports said.
The serviceman has been named as 23-year-old Hadar Goldin from Kfar Saba.
The Israeli military also claimed the apparent abduction occurred about 90 minutes into a 72-hour cease-fire that had been announced by United Nations Secretary-General Ban Ki-moon and US Secretary of State John Kerry late Thursday.
Meanwhile, only few hours after a cease-fire went into effect, Israeli shelling near the southern Gaza town of Rafah killed at least 40 Palestinians on Friday, the local hospital said.