Israeli Prime Minister Benjamin Netanyahu said he had ordered the military to prepare for a "significant expansion" of the ground offensive.
"It is not possible to deal with tunnels only from the air. It needs to be done also from the ground," he told a special Cabinet meeting in Tel Aviv. "We chose to begin this operation after the other options were exhausted and with the understanding that without the operation, the price we will pay can be very high."
Frustrated by Hamas' refusal to accept an Egyptian-brokered truce agreement and the failure of a 10-day campaign of more than 2,000 airstrikes to halt relentless rocket fire on Israeli cities, Israel launched a ground offensive it had previously been reticent to undertake to further weaken Hamas militarily.
With calls from Israeli hard-liners to completely crush Hamas, it remains unclear how far Israel will go in an operation that has already seen 274 Palestinians killed in Gaza, a fifth of them children.
"It won't end that quickly," said Yitzhak Aharonovitch, Israel's minister of public security. "Anything can happen. If we need to keep going, we will keep going. We won't stop. We need quiet for the citizens of the south and the citizens of Israel."
In a fresh effort to broker a truce, U.N. chief Ban Ki-moon was to leave Saturday for the Middle East to help mediate the Gaza conflict, Under-Secretary-General for Political Affairs Jeffrey Feltman said in New York. He spoke at a Security Council emergency meeting on the conflict Friday, adding that a ceasefire remains "indispensable" for urgently needed humanitarian efforts to succeed.
The Israeli military said it had killed nearly 20 militants in exchanges of fire since the ground offensive started on Thursday night.
Gaza health officials said more than 30 Palestinians have been killed since then, including three young siblings from the Abu Musallam family who were killed when a tank shell hit their home.
At the morgue, 11-year-old Ahmed's face was blackened by soot, and he and his 14-year-old sister, Walaa, and 16-year-old brother, Mohammed, were wrapped in white burial shrouds. Their father, Ismail, said the three were sleeping when the shell struck and he had to dig them out from under the rubble.
Israel says it is going to great lengths to avoid civilian casualties and blames them on Hamas, accusing it of firing from within residential neighborhoods and using its civilians as "human shields." On Thursday, the U.N. refugee agency for Palestinians, UNRWA, said a routine check in one of its vacant Gaza schools found about 20 hidden rockets and called on militants to respect the "sanctity and integrity" of U.N. property.
Critics say it is the intense fire itself in such a densely populated area that leads to the deaths of innocent civilians. The U.N. children's agency, UNICEF, said at least 59 or one in five of the Palestinians killed children were under the age of 18. UNRWA said 40,000 Palestinians were seeking refuge in 34 of its shelters throughout the Gaza Strip.
Most countries have expressed support for Israel's right to defend itself, while urging it to minimize civilian deaths in its ground assault. President Barack Obama spoke with Netanyahu on Friday and expressed his concern "about the risks of further escalation and the loss of more innocent life."
The operation also brought Israel its first military casualty. The circumstances behind the death of Staff Sgt. Eitan Barak, 20, were not made clear: Hamas's military wing said it ambushed Israeli units in the northern town of Beit Lahiya, but Israeli media said Barak was likely killed by friendly fire. The army said a number of soldiers were also wounded. Earlier in the week, an Israeli civilian died from Palestinian mortar fire and several others have been wounded.
"The ground offensive does not scare us and we pledge to drown the occupation army in Gaza mud," Hamas spokesman Sami Abu Zuhri said in a statement.
Israeli public opinion strongly supports the offensive after days of unrelenting rocket fire from Gaza and years of southern Israeli residents living under the threat. Gaza militants have fired more than 1,500 rockets at Israel over the past 11 days, and rocket fire continued across Israel Friday.
The order to launch the ground operation was triggered not by the rocket fire, but by a Hamas attempt to infiltrate Israel on Thursday, when 13 armed militants sneaked through a tunnel from Gaza and were killed by an airstrike as they emerged inside Israel.
The military, which has already mobilized more than 50,000 reservists, said paratroopers had uncovered eight tunnel access points across the Gaza Strip and engaged in several gun battles with Hamas militants who ambushed them.
Israeli forces are expected to spend a day or two staking ground within two miles (three kilometers) of the border in the north, east and south of the Gaza Strip. Then, they are expected to begin destroying tunnels, an operation that could take up to two weeks. Tanks, infantry and engineering forces were operating inside Gaza, where the military said it targeted rocket launchers, tunnels and more than 100 other targets.
Hamas has survived Israeli offensives in the past, including a major three-week ground operation in January 2009 and another weeklong air offensive in 2012, but in each case the militant group recovered. It now controls an arsenal of thousands of rockets, some long range and powerful, and it has built a system of underground bunkers.
But Hamas is weaker than it was during the previous two offensives, with little international or even regional support from its main allies, Turkey and Qatar. Protests against the offensive took place Friday in Turkey, Jordan and the West Bank.
Egypt, which has been pushing for a cease-fire, is at odds with Hamas' conditions, which include a lifting of the siege of Gaza and completely open borders into the Sinai where Egypt is already fighting Islamic extremists.