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.

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 criticism from Egypt.

(Reporting by James Mackenzie in Jerusalem, Nafisa Eltahir in Cairo and Jana Choukeir in Dubai; Editing by Alison Williams)