Egypt also wants to see a defined time frame for the talks.
"Any negotiation, for which a basis and a goal is agreed, must have a time-frame," Abul Gheit said.
The two leaders also discussed a prisoner swap between Israel and the Palestinians that would see the release of hundreds of Palestinian prisoners in exchange for Gilad Shalit, an Israeli soldier captured in Gaza three and half years ago.
Abul Gheit said the deal was still "suspended" and that a Hamas delegation currently in Syria was to head back to Cairo for talks with officials on the issue.
Fahmi Huwaidi, an Egyptian political analyst, told Al Jazeera that a deal on a prisoner exchange between Israel and Hamas could be reached during the visit.
"Israel seeks by this visit to confirm the importance [it] gives to Egypt's role in this matter, particularly after Germany began to mediate.
"The Israeli prime minister seeks to gain concessions from Hamas in the prisoners exchange deal, after the pressure currently being put on the movement.
"This visit is also an attempt by an ally to save his ally, after the criticism against the Egyptian leadership concerning the building of a steel wall on the Egypt-Gaza border."
Egypt has been criticised for keeping closed its border with the Gaza Strip, the only Gazan border not affected by an Israeli blockade, and building a wall along the divide.
The US administration of Barack Obama, the US president, is said to be drafting letters of guarantee for Israel and the Palestinians to serve as a basis for the relaunch of peace talks, stalled for almost a year.
"The mere fact of Mubarak meeting Netanyahu in Egypt is in and of itself an instrument of pressure on Mahmoud Abbas [the Palestinian president]"
senior fellow of the Institute of Palestinian Studies
"We've been hearing increasing reports that the Americans may be about to unveil a new diplomatic initiative," Mouin Rabbani, a senior fellow at the Institute of Palestinian Studies, told Al Jazeera.
"The mere fact of Mubarak meeting Netanyahu in Egypt is in and of itself an instrument of pressure on Mahmoud Abbas [the Palestinian president] to accept whatever conditions the Americans may be putting forth.
"The message that is coming from the Egyptian leadership now is that what the Americans are going to propose is good enough for the most important Arab state and therefore it should be good enough for the Palestinian leadership," he said.
One Arab diplomat in Cairo was quoted by the AFP news agency as saying that George Mitchell, the US special envoy to the Middle East "will present two draft letters of guarantee, one for Israel and one to the Palestinian Authority during his next visit to the region".
Egypt had already asked for written US guarantees before peace negotiations could resume, in order to ensure that their aim is the establishment of a Palestinian state within 1967 borders.
But peace efforts have already been overshadowed by Israel's announcement on Sunday that it had invited tenders for the construction of hundreds of new homes for Jewish settlers in occupied Arab East Jerusalem.
The announcement prompted criticism from the US and EU, alongside Egypt.
Netanyahu announced a 10-month moratorium on new housing projects in Jewish settlements in the occupied West Bank in November, but that suspension was not applied to East Jerusalem, which was annexed by Israel in 1967.
Separately, four Egyptians filed a request with the country's prosecutor general demanding Netanyahu be arrest over his treatment of the Palestinians.
In their request, the activists stated they were "filing this on grounds that the constitution states that Egypt is part of the Arab nation ... and that the Palestinian issue is the central Arab cause".