Menahem Kahana/Pool/AP
Israeli Prime Minister Benjamin Netanyahu chairs a cabinet meeting in Tel Aviv on Sunday.
CNN  — 

Israel has proposed a fresh pause in fighting in exchange for Hamas releasing hostages that are still held captive in Gaza, a source familiar with the negotiations told CNN.

The development would mark a significant shift since Israel’s military operations resumed in the war and formal hostage negotiations came to a halt.

Adding to hopes of a new cessation in fighting, Hamas’s political chief visited key mediator Egypt on Wednesday.

The general framework that Israel is proposing would once again entail hostages – women included – being released in phases, with the ultimate goal of all hostages in Gaza getting out.

But the source stressed that a deal at this moment is not imminent, and that the discussion of proposals mark, at most, the beginning of negotiations. Still, Israel appearing to return back to the negotiating table is notable.

Prime Minister Benjamin Netanyahu and his government face domestic pressure over hostages, particularly after the Israel Defense Forces (IDF) mistakenly killed three Israeli hostages.

Another source familiar with the discussions said that a proposal Israel has made to Hamas, via mediator Qatar, is that fighting be paused for a week in order to allow for the release of hostages.

Amir Levy/Getty Images
A wall in Tel Aviv is covered with images of hostages held in Gaza.

Axios was first to report that Israel is offering a weeklong pause in the fighting as part of the negotiations.

It comes as Ismail Haniyeh, the head of Hamas’ political bureau, arrived in Cairo for talks with Egyptian officials on the latest developments in the Israel-Gaza conflict, Hamas said in a statement on Telegram Wednesday.

Hamas, too, is under significant pressure, and both US and Israeli officials have said they believe the resumption of fighting, and particularly the IDF targeting southern Gaza, would ramp up the pressure on Hamas to accept another temporary ceasefire.

Hamas is asking for “more heavy-duty prisoners than before,” an Israeli official said, a reference to the release of Palestinian women and teenaged prisoners from Israeli jails in exchange for hostages in the last deal between the two sides.

The deal last month was the result of painstaking negotiations between Israel, Hamas and the US, with Qatar playing a major mediating role. Beginning on November 24, the deal was renewed twice before ending on Friday, December 1.

During the first seven-day pause in fighting, Hamas had to release each day at least 10 Israeli hostages kidnapped on October 7. The truce came to an end when Hamas refused to release the remaining female hostages. Since then, US and Qatari officials have continued to discuss ways to secure the release of additional hostages.

The framework saw 80 Israelis, some of whom hold dual citizenships, released from captivity. Additionally, a number of foreign nationals — 23 Thai citizens, one Filipino and one dual Israeli-Russian citizen — were freed as part of separate negotiations outside the truce.

On the other side, a total of 240 Palestinians were freed from Israeli prisons, mainly women and minors, and many of whom had been detained but never charged.