According to Reuters, Islamic State claimed responsibility for the incident 800 metres from the entrance to St Catherine's Monastery.
It also left four other police officers injured.
The attack comes days after 45 people were killed in bomb attacks which struck Coptic churches in Tanta and Alexandria on Palm Sunday, and ten days before Pope Francis is due to visit Egypt.
The country's Christian minority has increasingly been targeted by Islamist militants.
An Islamist insurgency has gained pace in northern Sinai since President Mohamed Mursi of the Muslim Brotherhood was overthrown as president in 2013.
By contrast, such incidents are less common in southern Sinai, which is a popular tourist destination.
St Catherine's Monastery is part of the Eastern Orthodox Church.
The attack nearby comes ahead of a decision by Russia on whether to resume flights to Sharm el-Sheikh after a Russian airliner was downed in 2015.