Tel Aviv, Israel - As the Israeli army launched its first ground incursion into the Gaza Strip overnight on Sunday, and Israeli air strikes on the territory continue, talk of a mediated ceasefire between Israel and Hamas remains mired in uncertainty.

The Israeli newspaper Yediot Aharonot reported on Saturday that Egypt was talking with another Arab state, "most probably Qatar", about the terms of a ceasefire. Tony Blair, the Quartet envoy, arrived in Cairo on Saturday and met with Egyptian President Abdel Fattah el-Sisi, who warned against "further escalation" in Gaza.

But Egyptian and Palestinian sources have said that previous attempts at mediation have ended in failure.

"There were some initial attempts to calm things down, but the message from Hamas is that there’s no real mediation," said Issandr El Amrani, the North Africa director at the International Crisis Group. "The players are not ready to do it."

US President Barack Obama spoke with Israeli Prime Minister Benjamin Netanyahu on Thursday and offered to help broker a truce, though the State Department clarified that it would not talk directly with Hamas, which Washington considers a terrorist organisation.

His message was undermined on Friday, though, when Dan Shapiro, the US ambassador in Tel Aviv, told Israeli Army Radio that Israel would enjoy American support "even if it decides" to launch a ground invasion.

Israeli officials maintain that any ceasefire is "up to Hamas", and that they would be satisfied with a return to the ceasefire which ended Israel's 2012 military offensive. But any concessions to Hamas will be a tough sell within Israel, after weeks of dire talk from officials about the need to destroy the group.

Netanyahu, notably, toned down his rhetoric in a press conference on Friday evening, speaking only of the need to stop the rocket fire. "The military strikes will continue until we can be certain that the quiet has returned to Israeli citizens," Netanyahu said. "I would like to make it clear that no terrorist target in the Gaza Strip is immune."

INTERACTIVE: #GazaUnderAttack

In Gaza, Palestinians have demanded an end to Israel's aerial bombings, as more than 165 Palestinians have been killed and nearly 1,000 more injured in six days of attacks.

Sami Abu Zuhri, a Hamas spokesperson, told Al Jazeera that the group is demanding the release of Palestinian prisoners that were freed in a prisoner swap in 2011, but were subsequently rearrested. But Naftali Bennett, the Israeli economy minister and a key member of Netanyahu's coalition government, said last month that his party would oppose any more prisoner deals. "After 30 years, it's clear that Israel should not release any more terrorists, in any situation, period," he said.

Other Hamas prerequisites to reaching a truce are for Israel to stop its attacks on Gaza, the West Bank and Jerusalem, and for Israel to abide by the 2012 ceasefire agreement, Abu Zuhri said.

[Israel] started the fight. We will remain on our land and the future is for us. If there [is] a lull, this will not be the last battle, but a warrior's respite. - Moussa Abu Marzouk, Hamas

The Israeli daily Haaretz reported on Sunday that the ceasefire terms are "more-or-less set", and include a one-year period of calm on both sides, and possibly, the presence of Palestinian Authority security forces in the Gaza Strip, manning the Rafah crossing with Egypt in coordination with European Union inspectors.

"The [talk] about efforts to broker a ceasefire come from Israeli sources only," Abu Zuhri told Al Jazeera, adding that media reports of a ceasefire are meant to distract international attention from the violence currently engulfing Gaza.

The deputy secretary-general of Palestinian Islamic Jihad, Ziyad al-Nakhale, told the London-based Al-Hayat newspaper that the 2012 ceasefire was behind them because "the Israelis did not honour it". He also said that there have been no concrete discussions on reaching a halt to the hostilities.

But Moussa Abu Marzouk, a member of the Hamas leadership, recently hinted at a ceasefire on his Facebook page. "[Israel] started the fight. We will remain on our land and the future is for us. If there [is] a lull, this will not be the last battle, but a warrior's respite," he wrote.