Israeli Defence Minister Ehud Barak warned that a flare-up in violence with Gaza was "not over," after new rocket fire in the morning and Israeli strikes overnight.
The violence that began on Saturday appeared to have slowed considerably, with Gaza militants firing a single rocket into Israel on Tuesday morning, hours after they said they would commit to a ceasefire if Israel state did the same.
Israeli war planes carried out air strikes against several targets overnight, which caused no injuries, although medics in Gaza said Tuesday a seventh person had died in the violence, succumbing to wounds he sustained on Saturday.
Barak, after meeting Israeli military chiefs on Tuesday morning, warned that the current round of confrontations was ongoing, adding that Israel would decide how and when to respond to the rocket fire.
"It is certainly not over and we will decide how and when to act if necessary," he said in remarks communicated by his office.
The main Palestinian fighter groups in the Gaza Strip said on Monday that they are ready for a ceasefire if Israel "stops its aggression" against the territory.
"The Islamic and nationalist movement confirm that the response of the resistance depends on whether the Zionist aggression against our people is continued," they said in a statement read out at a news conference in Gaza City.
The offer was made after fighters fired more rockets at southern Israel, wounding eight people, in retaliation for the killing of seven Palestinians on Saturday.
On Sunday night, Israel carried out air strikes in northern and southern Gaza, and Palestinians began launching rockets into Israel from around 7:30am local time on Monday morning.
'Act to stop it'
Israel's top military officials were said to be weighing whether to launch a more serious response to the violence, and Defence Minister Ehud Barak warned that efforts to stamp out rocket fire would intensify.
"These are very important days ... in light of the ongoing activity against Hamas and terror organisations in Gaza, which is likely to intensify and expand," he said, quoted by his office.
Prime Minister Binyamin Netanyahu, touring southern Israel with foreign diplomats, warned that "the world must understand that Israel has the complete right and the obligation to protect its citizens".
"We shall not sit with our arms folded in the face of repeated, almost daily, attacks on our citizens," he added. "We shall act to stop it."
US ambassador Dan Shapiro, writing on Facebook, has said "the United States supports Israel's right to defend itself and its citizens from these attacks."
In light of the violence, European Union foreign policy chief Catherine Ashton on Monday urged both sides to "refrain from exacerbating the situation."