The Israeli military has threatened to invade Gaza amid warnings that its bombing of the territory was already leading to war.
The UN's special envoy to the Middle East peace process has called for a ceasefire, saying "leaders on all sides have to take the responsibility of de-escalation."
Israeli prime minister Benjamin Netanyahu warned: "Those who attack us have signed their death warrants."
Hamas was reported to have asked for a ceasefire but continued to fire rockets into Israel after days on unrest that has continued to intensify.
The US sent an envoy to Israel to seek a reduction in violence.
President Joe Biden, speaking after a phone call with Mr Netanyahu, said: "Israel has a right to defend itself against thousands of rockets flying into their territory.
He added: "My hope is that we'll see this coming to a conclusion sooner rather than later."
Israeli jets demolished a series of high-rise tower blocks with airstrikes in Gaza, while Israeli forces ordered tanks and troops to the border.
Video: At least 53 Palestinians and six Israelis killed since Monday (France 24)
At least 53 Palestinians and six Israelis killed since Monday
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Continued strikes on both sides brought the death toll in Gaza to 53, including 14 children.
Israel showed no sign of complying with calls for a ceasefire.
Its military spokesmen said they were preparing for all eventualities. This included an incursion into the Gaza strip, it was reported.
The Israel Defence Forces, meanwhile, said their "marching orders" were to intensify operations.
Hamas carried out three attacks on Tuesday night, attempting to overwhelm the Israeli Iron Dome defence system. Two volleys of hundreds of rockets were aimed at Tel Aviv.
At least 100 of the more than 1,000 rockets fired from Gaza have got through, some hitting cars and buildings.
A soldier was killed when his vehicle was struck by an anti-tank missile fired from Gaza.
Israel responded in force. On Tuesday, the Hanadi Tower, the highest building in the Gaza strip, was razed.