Israeli military said the incidents took place during a security operation against smugglers in the Naqab (Negev) desert.
At least three Israeli soldiers have been killed in rare shootings near the Egyptian border, and a fourth one injured, the army has said.
The exchange of fire on Saturday reportedly took place around the Nitzana/al-Awja border crossing between Israel and Egypt, which is located about 40km (25 miles) southeast of the point where Israel’s border with Egypt and the Gaza Strip converge. It is used to import goods from Egypt destined for Israel or the Gaza Strip.
According to Israeli media, two of the soldiers were found dead at an Israeli military guard post after they failed to respond to their radios, about two hours after arriving at the post for their shift.
The Israeli army said the first exchange of fire took place on Saturday morning. The second border clash occurred at noon, in which a third Israeli soldier was killed.
The Israeli army announced it had killed the gunman in the second exchange of fire, who was reportedly dressed in the uniform of an Egyptian border guard and had infiltrated the border where the army said it was operating to thwart smuggling activity.
“During an encounter with an assailant in Israeli territory, a short while ago, an exchange of fire developed. Troops and commanders engaged [the suspect] and shot and killed him,” the army said.
“This morning, a security incident occurred in the area of the Paran Regional Brigade. Two [Israeli] soldiers, a man and a woman, were killed by live fire adjacent to the border,” the military said in a statement, adding that the incident is under review.
The army had initially declined to confirm the deaths, which were reported by Israeli media, while families were notified.
According to Al Jazeera’s bureau chief in Palestine, Walid al-Omari, the last time a smuggler attempted to infiltrate the Egyptian-Israeli border was a year and a half ago, when he was killed by Israeli forces.
Al-Omari added that the border was active between 2004 and 2012 until Israel built a 241km border fence that extends from Eilat and Taba on the Red Sea to the meeting point of the Egyptian-Israeli-Palestinian border east of Rafah.
The fence was to stop the entry of African refugees and migrants, and armed fighters linked to ISIL (ISIS) who are active in Egypt’s Sinai desert.
Egypt was the first Arab country to sign a peace treaty with Israel in 1979 and maintain close security ties. Fighting along their shared border is rare.