Earlier, Israeli strikes killed at least six Palestinians in the Gaza Strip on Thursday, including two children, medics said.
A Hamas official and a diplomat close to the Egypian talks said discussions were "positive" and "making progress".
The official said the progress would be tested by Israel's response and whether this allowed a broad accord to be reached.
However, the Israeli official, who declined to be identified, said Israel had rejected a proposal.
"Israel demands the complete disarmament of Hamas, including the removal of weapons from Gaza and the complete demilitarisation of the Strip as a precondition for any process," the official said.
"The 15-point document in question does not provide a satisfactory response to these demands and Israel conveyed its objections."
Russian deputy ambassador to the United Nations Anna Evstigneeva told the Security Council on Tuesday that a road map prepared by US President Donald Trump's Board of Peace envoy for Gaza Nikolay Mladenov had so far failed to gain the support of Israel and Hamas, with the demilitarisation of the enclave remaining the main stumbling block.
The diplomat close to the talks said the negotiations were making progress toward a road map that would result in all heavy and light weapons in the Gaza Strip being decommissioned.
Hamas leaders are holding talks with mediators from Egypt, Qatar and Turkey on the second phase of Trump's plan, as presented to them by his Board of Peace, which oversees its implementation.
The plan calls for a surge in humanitarian aid, governance by a civilian Palestinian administration, the disarmament of Hamas, the withdrawal of Israeli forces from the strip and the deployment of an international force to help maintain security.
But progress has stalled in previous months.
On Thursday, a Hamas official said the group was coming with a "positive and good" response but did not say whether the group had agreed to fully disarm - a key sticking point in the talks over the past four months.
Hamas wants Israel to commit to ending attacks on the Gaza Strip and withdrawing its forces.
Sources close to Hamas said the group was willing to "confine and store" heavy weapons under a Palestinian body, not hand them over to Israel.
Hamas did not comment on the matter.
It is not yet clear what will happen regarding light or personal weapons or whether the new Hamas stance will be accepted by Israel or the Board of Peace.
The diplomat involved in the talks said the road map under discussion in Cairo would also include authority being transferred in phases to the US-backed National Committee for the Administration of Gaza (NCAG), under the principle of "one authority, one law, one weapon".
The plan includes the elimination of tunnels, arms stockpiles and weapons production facilities, the diplomat said, adding that by the end of the process there would be no remaining militant infrastructure in the Gaza Strip.
The diplomat described the road map as being based on reciprocal steps rather than trust.
So far, the reality on the ground in Gaza remains far from this road map.
The Israeli military has continued to expand its military occupation of the Gaza Strip, with Prime Minister Benjamin Netanyahu saying that Israel intends to expand the area under its control to 70 per cent of the enclave.
It remains unclear how many forces would be available to deploy with the planned International Stabilisation Force, and whether Israeli troops would withdraw from the enclave.
In Cairo, two Egyptian sources said Hamas had agreed to several contentious points, including the handover of weapons, but had requested changes regarding the definition of the infrastructure to be dismantled, adding more talks were underway.
