The death in the early hours of Sunday morning occurred as police investigated the bomb attacks in the resort of Dahab and north Sinai, security sources said.
Two people involved in the gunfight escaped, the sources added.
However, separate security sources said that one man was killed and no one was captured. Details of the shootout were not immediately available.
Bomb attacks killed 18 people in Dahab on Monday, and on Wednesday two men died after blowing themselves up in attacks on a multinational peace force and a police station, both based in the north of Egypt's Sinai peninsula.
On Saturday police detained two men they said drove people suspected of involvement in the attacks. Shortly after the Dahab blasts, police said they had formally detained at least 10 people and taken in about 70 local beduin for questioning.