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Pushes for a ceasefire and a hostage release deal in Gaza have increased in recent days, ahead of the holy Muslim month of Ramadan due to start on Monday, US news outlets have reported.

US, Qatari and Egyptian mediators have held talks with each other throughout the week, with CIA chief William J. Burns speaking with Egyptian intelligence chief Abbas Kamel and Qatari Prime Minister Sheikh Mohammed bin Abdulrahman al-Thani.

Burns also held a secret meeting with Mossad chief David Barnea in Jordan, Benjamin Netanyahu’s office confirmed.

Following the meeting, an Israeli statement claimed: "At this stage, Hamas is entrenching its positions like someone who is not interested in a deal and is striving to inflame the region during Ramadan at the expense of Palestinian residents of the Gaza Strip".

It remains unclear if any progress was made following the talks, before Burns returned to the US on Saturday, the Washington Post said. The US website said that the issue of a temporary pause versus a permanent end to the fighting led to last weekend’s breakdown of talks held in Cairo.

Israel refused to send a delegation last week during the meeting, which yielded little to no results.

The US and Israel have urged Hamas to accept their ceasefire proposal, which would include a six-week truce and the release of at least half of the remaining hostages held in the enclave, estimated to number at 100.

Hamas, however, has called for a permanent ceasefire and Israeli withdrawal from the devastated enclave, where at least 31,045 Palestinians have been killed since October 7, amid a brutal and indiscriminate Israeli land and air offensive.

Israel Prime Minister Benjamin Netanyahu has reportedly refused such conditions, and has vowed to "completely destroy" Hamas throughout the war.

Hamas has also made demands for a specific amount of aid to enter the Gaza Strip, as well as the return of displaced Palestinians to the north. Israel has however demanded to retain freedom of military movement in the much-battered north, and has requested complete list of the remaining hostages and how many of them are still alive.

Hamas says it is being flexible in trying to reach an agreement withÌýIsrael via mediatorsÌýfor a ceasefire inÌýGaza, and accused Israel of evading parts of an agreement that would lead to a permanent ceasefire.

The meetings came amid concerns about tensions in the occupied West Bank and Jerusalem during Ramadan, if a truce isn’t put into effect before the holy month begins.

Israeli extremists and police often attack Palestinian worshippers at the Al-Aqsa Mosque during Ramadan, with Israeli attempts to restrict the number of those attending prayers at the compound.

Abu Obeida, the spokesperson for Hamas’ armed wing has on Friday called for "confrontation and demonstration, on all fronts inside and outside Palestine" and for Palestinians to "mobilise towards al-Aqsa mosque," during Ramadan.

Israel has said that Muslim worshippers will be allowed to enter the Al-Aqsa compound in "similar numbers" as in recent years, but that this will be followed by "a weekly situation assessment". Israel however has a track record of limiting and barring worshippers from entering.

The US administration of President Joe Biden is under increasing pressure to secure a deal immediately. The US reportedly envisions what a senior administration official called a "three-phase deal", the Washington Post reported.

The deal would begin with an initial six-week pause in fighting, the concurrent release of a number of hostages, and a guaranteed increase in humanitarian relief.Ìý

The US hopes that the initial six-week truce would lead to something more permanent, the official said.

Axios reported that the US is determined that a truce deal is reached in a bid to salvage the broader US strategy in the region, which includes Saudi normalisation with Israel, which was halted after Israel launched its brutal attack on Gaza.

US ambassador to Israel Jack Lew stressed that Washington is doing everything it can to "keep the conversation going" around the hostage and truce deal,ÌýAxios added.

US and Qatari officials, including White House National Security Advisor Jake Sullivan, said that both countries will push for a ceasefire deal during Ramadan, which has been confirmed to start on Monday in much of the Muslim world following the sighting of the crescent moon by Saudi Arabia.

However, people in Gaza are likely to continue to suffer from Israeli aggression and blockade throughout Ramadan, when families traditionally gather to break the fast at sunset.Ìý A looming famine still threatens the enclave, with at least 25 people dying due to malnutrition and dehydration triggered by insufficient food, water and medicine.