Hamas' military wing said Nour Baraka, a Qassam Brigades commander, died in the fighting.
A Israel Defense Forces spokesperson said the shooting broke out "during security activity by the IDF in the Gaza area."
Palestinian sources told Haaretz the IDF also executed drone strikes in the area at the same time as the shooting. Qassam Brigades said the IDF used the cover of airstrikes to retreat back to the Israeli side of the border.
The IDF said no Israeli soldiers were kidnapped during the fighting, but offered no details about casualties to its forces.
Israeli Prime Minister Benjamin Netanyahu said he would cut short his trip to Paris for World War I commemorations and return to Israel.