Asked about the reports, an Israeli military spokesman said she could not comment on troop deployments.
Israel has signalled it is winding down the 25-day-old Gaza war unilaterally, saying it will not attend Egyptian-hosted negotiations for a new truce and giving Palestinians who had fled fighting in one northern town the all-clear to return.
Shelling exchanges continued, pushing the Gaza death toll given by Palestinian officials up to 1,669, but in some areas witnesses reported Israeli tanks pulling back toward the border.
Israel said Palestinians launched 74 rockets across the border, most of which fell harmlessly wide while seven were shot down by its Iron Dome interceptor, including over Tel Aviv.
Several ceasefires between Israel and the Gaza Strip's dominant Islamist Hamas faction had failed to take hold or quickly collapsed, most recently on Friday after two Israeli soldiers were killed and a third went missing in an ambush.
Israel accused Hamas of seizing Lieutenant Hadar Goldin and the United States blamed the group for a "barbaric" breach of the truce.
The United Nations was more guarded in its censure of Hamas but demanded Goldin's release.
Seeking to shift responsibility, Hamas said it believed its gunmen had struck before Friday's ceasefire began and that if they captured Goldin, he probably died with his captors in heavy Israeli barrages that followed.
A Palestinian delegation arrived in Cairo for new truce talks, which would include Hamas's demand Egypt ease movement across its border with blockaded Gaza.
Turning its back on those negotiations, Israel said it would not send envoys as scheduled.
"They (Hamas) cannot be trusted to keep their word. They cannot stop (firing) because, for them, a ceasefire at this stage, whether by arrangement or not by arrangement, would mean acknowledging the worst possible defeat," Deputy Foreign Minister Tzachi Hanegbi told Israel's Channel Two TV.
"I believe this is the point at which the ground manoeuvres should be brought to an end. Hamas can be hit as much as will be required in response to firing that, I expect, will persist."
A poll by Israel's Channel Ten TV found 32 percent of the public wanted a truce, 31 percent wanted ground forces withdrawn even without a halt to hostilities, and 31 percent wanted the military to step up operations, reoccupy Gaza and topple Hamas.
Hamas, its guerrillas massively outgunned by a Jewish state it considers an eternal enemy, said it would prevail.
Any unilateral pullout by Israel would mean "it has failed to achieve any of its goals and would be a clear defeat for the occupation army and for its leaders," Hamas's bloc in the Palestinian parliament said in a statement.
"Gaza resisted, endured and will achieve victory."
Israel launched a Gaza air and naval offensive on July 8 following a surge of cross-border rocket salvoes by Hamas and other Palestinian guerrillas.
With US backing, Israel had said that whether or not there is a ceasefire its forces would pursue their main mission of hunting tunnels used by Hamas for several cross-border attacks.
More than 30 of these, and dozens of access shafts, have already been unearthed and were being blown up, the military said.
"Our understanding is that our objectives, most importantly the destruction of the tunnels, are close to completion," a military spokesman, Lieutenant-Colonel Peter Lerner, said.
Crowded Gazan towns close to the Israeli border have seen devastating clashes, and the flight of tens of thousands of Palestinians, as tanks and troops swept in to confront dug-in guerrillas after the army warned civilians to evacuate.
The Palestinian Centre for Human Rights said 520,000 people had been displaced by the fighting - more than a quarter of Gaza's population.
Another group, the Al-Mezan Centre for Human Rights, said some 3,000 homes had been totally or partly razed.
Israel said on Saturday evacuees from Beit Lahiya, a northern town of 70,000 residents, could return.
But fear still gripped the townspeople.
"No one has told us to go back," said Talab Manna, a 30-year-old father of seven camped out at a UN-run school serving as a refugee haven.
"We can't risk going back and being bombed by the Israeli forces."
After Friday's ceasefire was shattered, Israeli Prime Minister Benjamin Netanyahu called his security cabinet into special session to discuss Goldin's suspected capture in the southern town of Rafah, and warned Hamas and other Palestinian guerrillas would "bear the consequences of their actions".
But Israeli media said Netanyahu and Defence Minister Moshe Yaalon would likely hold course in Gaza rather than escalate.
Citing an internal investigation complicated by inability to communicate with its gunmen in the Rafah area, Hamas's armed wing said on Saturday it believed their ambush happened an hour before the truce began, in response to Israeli troop advances.
Hamas said it did not know what had happened to the soldier but if he was captured, he probably died in Israeli hostilities that followed the ambush.
Quoting a senior military officer, Israel Radio also said Goldin's condition was not known.
It said he was last seen next to the two troops killed by a Hamas suicide bomber - suggesting Goldin may not have survived and his captors had a corpse.
Israel has confirmed that 63 soldiers have died in combat.
Palestinian shelling has also killed three civilians in Israel.
Hamas had long threatened to capture Israelis for a prisoner swap.
In 2011, Israel released more than 1,000 Palestinian prisoners in exchange for Gilad Shalit, a soldier snatched by Hamas five years earlier.
Israel has twice freed prisoners for the bodies of soldiers held by Lebanon's Hezbollah militia.
Rafah in ruins
The Rafah clash triggered Israeli shelling from the middle of Friday morning that killed 150 Palestinians.
By afternoon, Israel declared an end to the truce - which was meant to have lasted 72 hours, allowing humanitarian relief to reach Gaza's 1.8 million Palestinians and for further de-escalation talks.
Medical officials on Saturday counted at least a dozen homes destroyed.
"It was like an action movie - explosions everywhere, cars flying up in flames, people crushed under houses that were bombed," local man Bassim Abed told Reuters.
Hamas said it launched long-range rockets on Saturday at the Israeli cities of Haifa and Tel Aviv.
There was no word in Israel of Haifa being struck, but the military said its Iron Dome interceptor had shot down rockets over Tel Aviv and the southern city of Beersheba.
No one was hurt by the salvo.
Among targets of Israeli air strikes on Saturday was a building in the Islamic University campus in Gaza City.
The military said the building had been used by Hamas for weapons research and development of weapon manufacturing, and that dozens of tunnels and arms caches had been bombed elsewhere.
"The Egyptian initiative is a real chance to find a real solution to the crisis taking place in the Gaza Strip," President Abdel Fattah al-Sisi told reporters.
"Lost time ... complicates the situation more and more."
Hamas is seeking an end to Israel's blockade of Gaza.
It also wants a hostile Egypt to ease restrictions at its Rafah crossing with the territory.
In a boost to Israel, the US Congress approved $225 million in emergency funding for Iron Dome.