Army spokesman Peter Lerner said in the last 24 hours "we have had at least six incidents where Hamas has fired at IDF activities," referring to the Israeli Defence Forces.
He said Israel had responded in what was the first direct confrontation between Israeli soldiers and Hamas fighters since the last major hostilities in Gaza ended in 2014.
Clashes between Palestinian fighters and IDF forces were continuing east of Rafah in the southern Gaza Strip this afternoon, security sources and witnesses said.
The flare-up has raised concerns over the fate of a ceasefire between Israel and Hamas that has held since the 50-day war left more than 2,251 Palestinians and 73 Israelis dead.
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Hamas's armed wing, the Ezzedine al-Qassam Brigades, accused Israeli forces of encroaching into Gazan territory and said it was ready to respond.
"We will not permit this aggression to continue," the group said.
Overnight Israeli air strikes hit four targets in Gaza, wounding four people, three of them children, medical and security sources said.
The army announced a new tunnel was found today, after a first one was discovered in mid-April.
"Overnight, in response to the ongoing attacks against Israeli forces, an Israel Air Force aircraft targeted four Hamas terrorist infrastructure sites in the northern Gaza Strip," the military said in a statement.
It gave no word on casualties but medical sources in Gaza said a raid on the Al-Zeitoun district wounded four members of the same family - three children and a 65-year-old man.
Lerner said the new tunnel discovered toay had been about 30 metres deep.
He did not say how far into Israel the tunnel stretched or when it was built, but confirmed the discovery was made by Israeli forces on the Gazan side of the border.