Talks on Sunday initiated by Qatar, the US and Egypt aimed at brokering a hostage deal between Israel and Hamas were “constructive” but meaningful gaps remain, a statement from the Israeli prime minister’s office has said. The statement said the parties would continue to hold discussions this week. “There are still significant gaps in which the parties will continue to discuss this week in additional mutual meetings,” the statement added. Before the talks in Paris, the Associated Press reported that US negotiators including the CIA director, William Burns, had provided a framework for negotiations focused on a two-month pause in fighting between Israeli forces and Hamas. US officials have reportedly proposed an initial 30-day temporary ceasefire to allow for the remaining female, elderly and wounded Israeli hostages to be freed. This would be followed by a second 30-day pause where Israeli soldiers and male hostages would be released, in tandem with an increase in the trickle of aid permitted into Gaza. The halt in fighting could, they hope, provide a further opportunity to negotiate a more durable, long-term ceasefire. During talks in Doha in recent weeks, Hamas has repeatedly declined to accept any deal that does not include a permanent ceasefire, according to reports. Israel recently offered a two-month pause to the fighting in exchange for hostages, but without guaranteeing a permanent end to the war. A source with knowledge of the discussions said the current proposal was to use a system of phased ceasefires and hostage releases to build confidence. If each stage was successful, they said, this could produce a clear and permanent end to the fighting. The location of the latest meeting means that Hamas negotiators cannot be present at the talks. Any advancements made in Paris would require the group’s political wing in Doha to convince leaders of its military wing, plus its top political official in Gaza, Yahya Sinwar, to agree to the deal. Officials from Hamas have repeatedly demanded a full ceasefire and the exchange of Israeli hostages for all of the thousands of Palestinian prisoners held in Israeli jails. Cairo previously proposed an ambitious deal for a 10-day halt in fighting in exchange for an Israeli withdrawal from Gaza, the release of hostages, and Hamas leaders’ exit from Gaza. The proposal appeared to fall flat, supplanted by small-scale talks to allow vital medications for the hostages into the enclave in exchange for a small increase in aid for Palestinian civilians. Egypt’s foreign ministry spokesperson, Ahmed Abu Zeid, said last week that Egypt is pushing for a ceasefire but Israel has imposed “unreasonable conditions” that have prevented this. Burns is expected to meet the head of Israel’s Mossad intelligence agency, David Barnea, the Qatari prime minister, Mohammed bin Abdulrahman al-Thani, and the Egyptian intelligence chief, Abbas Kamel, to discuss efforts to free the 136 remaining hostages held by Hamas and other militant groups in Gaza. The group met in Poland in late December in an attempt to restart negotiations to free the remaining hostages that had stalled after the release of 50 Israeli hostages in exchange for 180 Palestinian prisoners and a pause in fighting. In a separate deal 23 Thai hostages were freed after Palestinian militant groups took 250 people hostage when the group staged an unprecedented raid on Israeli territory on 7 October, killing 1,200 people. Israeli bombardments of the Gaza Strip in the past three months have killed more than 26,422 people, according to the latest estimates from the Palestinian health ministry in Gaza. The US president, Joe Biden, dispatched Brett McGurk, his envoy to the Middle East, to Cairo and Doha this week in an attempt to spur further progress on hostage negotiations. Biden also spoke with the Qatari emir, Sheikh Tamim bin Hamad al-Thani, and the Egyptian president, Abdel Fatah al-Sisi, on Friday night to boost efforts towards a new deal. Biden and Sisi agreed that “all efforts must now be made to conclude a deal that would result in the release of all hostages together with a prolonged humanitarian pause in the fighting”, the White House said in a statement. In Biden’s call with Thani, the two leaders “underscored the urgency of the situation and welcomed the close cooperation among their teams to advance recent discussions”, it added. Other White House officials preached caution. “We’re hopeful about progress, but … we should not expect any imminent developments,” said the national security council spokesperson, John Kirby, at a press conference. The meeting in France comes amid rising tensions across the Middle East, increasing the pressure on the White House and other negotiators to rapidly find a way to end the war in Gaza that they fear has sparked a widening conflict. Grant Shapps, the UK defence secretary, said on Sunday that British naval forces were “undaunted”, after HMS Diamond in the Red Sea shot down a Houthi drone, a day after an oil tanker managed by a company based in Britain was set alight while sailing in the Gulf of Aden. Talks in France also follow mounting pressure on the Israeli prime minister, Benjamin Netanyahu, to do more to free those held captive in Gaza, after 20 relatives of the hostages stormed Israel’s parliament this week to demand action to free their loved ones. Israelis have increasingly rallied to demand the prime minister’s resignation. Netanyahu has also faced criticism after leaked audio emerged in which he called Qatar’s role in mediation efforts “problematic”. Majed al-Ansari, a spokesperson for Qatar’s foreign ministry, said Doha was “appalled by the alleged remarks attributed to the Israeli prime minister in various media reports about Qatar’s mediation role”. “These remarks if validated, are irresponsible and destructive to the efforts to save innocent lives, but are not surprising.” If the remarks as reported were true, he added, Netanyahu “would only be obstructing and undermining the mediation process, for reasons that appear to serve his political career instead of prioritising saving innocent lives, including Israeli hostages”.