A member of Egypt's security forces was killed in a shooting incident near the Rafah border crossing between Egypt and the Gaza Strip and an investigation is under way, Egypt's military spokesperson said in a statement on Monday.

Israel's military had earlier said it was investigating reports of an exchange of fire between Israeli and Egyptian soldiers.

"A few hours ago (Monday), a shooting incident occurred on the Egyptian border. The incident is under review and discussions are being held with the Egyptians," the Israeli military said in a statement.

Later on Monday, Egypt warned against compromising the security and safety of its security forces, a high-level source told Egypt's state-linked Al Qahera News, adding that Egypt would take necessary measures to prevent the incident from recurring.

An Egyptian soldier stationed on a watchtower had reacted to seeing an armoured vehicle carrying Israeli troops cross a boundary line near the border while the soldiers pursued and killed several Palestinians, two Egyptian security sources said.

The soldier opened fire and Israeli forces fired back, killing him, the sources said, sparking an exchange of gunfire between the two sides. Several Israeli soldiers were wounded and the force fell back, and one Egyptian was wounded, the sources said.

Israel seized control of the Rafah border crossing from the Gaza side of the border earlier this month as it stepped up its military offensive in the area, drawing strong criticism from Egypt.

Egypt is concerned that Palestinians could be displaced from southern Gaza by the offensive, and has also repeatedly accused Israel of holding up deliveries of humanitarian aid to the territory, which Israel denies.

Egypt and Israel signed a peace treaty in 1979 and have cooperated closely on security issues around the borders between Israel, Egypt's Sinai Peninsula, and the Gaza Strip. They jointly upheld a blockade of Gaza after Hamas took control of the territory in 2007. (Reuters)