Col. Ahmed Mohamed Ali, an armed forces spokesman, said seven members of security forces and three passersby were injured Sunday in the battle with insurgents at a village, southeast of Sheikh Zawayed, the Egyptian newspaper al-Masry al-Youm reported.
Egyptian forces with 30 armored personnel carriers assisted by helicopters were involved in the fighting, Ma'an News Agency reported.
In related fighting, police headquarters in El-Arish were attacked Sunday morning, CNN reported.
Ali said one soldier was killed and eight injured in the clash with "criminal elements" using rocket-propelled grenades.
The incidents were the latest in a campaign seeking to find militants who killed 16 officers in an Aug. 5 attack on an army checkpoint on the border with the Gaza Strip.
Egypt closed the border crossing and the smuggling tunnels into Gaza following the attack, on suspicion that Palestinians were involved.
No Palestinians were found to be involved, a Palestinian official said Monday in announcing the findings of an investigation into the allegations, Ma'an reported.
Ehab al-Ghasein, an interior ministry spokesman, said no residents of the Gaza Strip or any other Palestinians played a part in the killing.
He cautioned against using any reports other than those issued by official state media as accurate accounts of the August attack.
Ghasein said some current reports of Palestinian involvement were designed to sow discord between Egypt and Palestinians.
In early September, Egypt said it had identified seven suspects, including one Egyptian, in the attack.
Four people whom officials said were wanted in the attack were detained Sunday in the Rafah village of Mahdia.