Some 4000 security personnel, bolstered by armoured vehicles, have been scouring the Sinai's desert plains and soaring mountains during the week searching for men involved in the 23 July Sharm el-Sheikh bombings, which killed at least 64 people, and the 7 October attack on a hotel in Taba and a nearby beach camp.
More than 30 died in those attacks.
Wednesday's shootout took place in four areas close to the Israeli border as police tried to storm hideouts of armed men suspected of having links to the Sinai bombings, at least two police officials said on condition of anonymity as they were unauthorised to speak to the media.
"There was strong resistance but the security forces are executing the operation as planned," said one of the officials.
Three security men were wounded on Wednesday in a landmine explosion in Qusaima, one of four mountainous areas surrounded by the forces about 20km west of the Israeli border, the official said.
The mine was believed to have been left in the area for at least two decades since the Arab-Israeli wars.
Another official said some 650 suspects have been arrested in Sinai since the search started on Monday.
Most of the arrests took place in el-Arish on the Mediterranean coast in northern Sinai.
"There was strong resistance but the security forces are executing the operation as planned"
Police statement on Wednesday's raid
In a separate clash on Tuesday, a police officer and two suspected fighters were wounded in a shootout east of the Suez Canal between security forces and bombing suspects.
Meanwhile, Egypt's state-run Al Ahram newspaper reported on Wednesday that police have arrested 20 people who might have had a direct links to the recent attacks.
Citing unidentified security officials, the paper said investigators believe they have netted the leaders of the cell that planned the attacks.
No further details were provided and officials were not immediately available for comment on the report.