An Israeli tank shell killed a member of the military wing of Gaza's ruling Hamas and wounded another on Thursday, Palestinian medical and security officials told AFP.
They said that Israeli aircraft also attacked an unknown target east of Gaza City, but there was no report of casualties.
The strike came after Hamas TV in Gaza said that three mortar shells were fired from the strip into southern Israel. It did not say who was responsible.
The Israeli army did not immediately confirm the air strike or any mortar attack.
The Palestinian officials named the dead man as Rabieh Barikeh and said that he and others were carrying out surveillance on Israeli movements at the Gaza border when they came under fire, east of Khan Younis in the central Gaza Strip.
The security officials and eyewitnesses said that an Israeli tank and armoured bulldozer came about 100 metres (yards) inside the strip.
An Israeli military spokeswoman told AFP that "a pinpoint and defensive activity against the terror tunnel is currently taking place."
She was referring to a tunnel from the Gaza Strip into Israel uncovered by Israeli security forces last month and allegedly intended as a springboard for attacks.
She gave no details of Thursday night's incident.
Hamas's Izzadine al-Qassam military wing said last month that it dug the tunnel as part of a plan to kidnap Israeli soldiers and hold them in exchange for Palestinians held in Israeli prisons.
"We are working on the ground and under the ground to release the prisoners (held by Israel)," the group's spokesman Abu Obeida said on October 20.
In June 2006 a group of Hamas and other militants snuck into Israel through a cross-border tunnel, seized Israeli soldier Gilad Shalit and took their prisoner back to Gaza the same way.
He was released on October 18, 2011 in exchange for 1,027 Palestinians held in Israeli prisons.
The trade was feted by Hamas as "an historic victory."
On Monday Israeli warplanes raided the northern Gaza Strip after militants fired across the border.
It was the first Israeli air strike on Gaza in more than two months and came shortly after two rockets were fired at southern Israel.
Witnesses said the air strike targeted a training ground west of Beit Lahiya that was used by Izzadine al-Qassam.
Hamas slammed the Israeli raid as "an escalation which aims to terrorise... our people," adding that it held Israel "completely responsible for any effects or deterioration."
Neither the air strike nor the earlier rocket attacks on southern Israel caused any damage or casualties.
The Israeli military said that the strike hit "two concealed rocket launchers" in northern Gaza.
On Sunday, a mortar shell was fired over the border that also caused no damage or injuries. In September, there were two instances of rocket fire, but Israel did not respond.
Monday's Israeli air strike was the first since August 14, when the air force hit targets in the same area, also in response to rocket fire.