Israel’s military says the two men behaved suspiciously and ignored warning shots.
The Islamic Jihad group said its fighters and members of another group had been operating near the fence when the exchange of fire occurred.
A member of the Democratic Front for the Liberation of Palestine was killed in the exchange, Islamic Jihad said.
Peretz allows army offensives
On Monday, Amir Peretz, Israel’s defence minister, authorised army operations inside Gaza against Palestinian fighters, despite a ceasefire agreement reached between Israel and the Palestinians in November.
Peretz said Israel would “not allow the continued strengthening and arming” of Palestinian groups in the coastal strip.
Some Palestinian groups have fired rockets into Israel despite the ceasefire.
Israeli security sources say the armed wing of Hamas has dug tunnels and upgraded its rocket arsenal in recent months.