Some signs of measured optimism emerged on Friday after talks in Cairo wrapped up the night before, as negotiators work to find a compromise on the deployment of IDF troops along the Gaza-Egypt border in the event of a hostage-ceasefire deal with Hamas.

The talks in Cairo on Thursday were “constructive,” an unnamed source familiar with the issue told Army Radio after Israel’s high-level negotiating team returned from Egypt.

According to the report, progress was made in closing gaps with Cairo over the Rafah Border Crossing, which has been shuttered since Israel occupied the Palestinian side of the Gaza-Egypt border in May.

“There has been progress made,” said a White House spokesman on Friday in a briefing with reporters.

Though a deal will ultimately have to be made between Israel and Hamas, discussions this week centered on finding a formula that Jerusalem and Cairo could agree upon. Prime Minister Benjamin Netanyahu insists that an Israeli presence on the Philadelphi Corridor — the buffer zone that separates Egypt from Israel and Gaza Strip — is vital to preventing Hamas from re-arming, while Hamas and Egypt want to see Israeli forces withdraw entirely.

The US has offered technological alternatives to an Israeli military presence on the border.

In his phone call with Netanyahu on Wednesday, US President Joe Biden asked the premier to pull Israeli forces back from a 1- to 2- kilometer section of the border road, according to Axios. The section is adjacent to the Tel al-Sultan refugee camp near the coast, where many Gazan refugees have taken shelter.

Citing three Israeli officials, Axios said that Netanyahu agreed to pull back from one IDF position. The official said that in response, Biden backed Netanyahu’s demand that Israeli troops continue to hold the rest of the border in the first stage of a deal.

Meanwhile, one of the prime minister’s advisors told Axios that Netanyahu agreed only to move one IDF position on the Philadelphi Route by several hundred meters.

Israeli and Arab media reported on Thursday that Israel came to Cairo with a new proposal that would see international forces on the Philadelphi Corridor, but the Prime Minister’s Office put out a statement denying that he would agree to such an arrangement.

Anonymous sources familiar with the negotiations told Ynet that progress was made in resolving disagreements between Jerusalem and Cairo on the future of the Philadephi Corridor.

After refusing to pass along to Hamas the US bridging offer presented by Secretary of State Antony Blinken in Doha last week, Egypt will reportedly pass the amended sections regarding the Rafah crossing and the Philadelphi Corridor to the terror group, Army Radio reported.

According to an Israeli official cited by Axios, once Biden backed Netanyahu’s proposed deployment, Cairo had no choice but to present the new maps to Hamas. However, Israeli officials do not believe Hamas will accept the updated formula.

The source quoted by Army Radio added that US and Egyptian negotiators are working around the clock to get the Palestinian terror group to participate in the next round of indirect negotiations, likely on Sunday.

The Israeli team in Cairo was led by Mossad chief David Barnea and Shin Bet chief Ronen Bar and included Maj. Gen. Eliezer Toledano, head of the IDF General Staff Strategy and Third-Circle Directorate, an Israeli official told The Times of Israel.

“There has been progress made,” said National Security Council spokesperson John Kirby. “We need now for both sides to come together and work toward implementation.”

Kirby added that meetings in Cairo are continuing Friday and that CIA Director Bill Burns will participate in those discussions.

Blinken said earlier this week that even if Hamas accepts the US proposal, the sides still will need to hold subsequent talks to finalize the implementation of the deal, adding another step to a process that has dragged on for months.

An Israeli official stressed to The Times of Israel on Thursday that “the prime minister stands behind the principle that Israel will rule the Philadelphi Route.”

The official said that “there is a reason we agreed” to the US bridging proposal offered in Doha last week, and that it meets Israel’s security demands.

Control over the Philadelphi Route is also the subject of debate within Netanyahu’s government.

His negotiators, said Channel 12 earlier this week, do not regard the prime minister’s insistence on an Israeli presence on the Philadelphi Corridor as essential, but rather as a “spoke in the wheels” of the talks. The IDF only moved to capture the Philadelphi Corridor in May, eight months into the war.

Meanwhile, hostages who had been released from Hamas captivity said Netanyahu promised them in a meeting on Friday that he would do everything he could to bring the rest of the abductees home, but not all were optimistic.

Ella Ben Ami, whose mother Raz Ben Ami was freed from Hamas captivity during a week-long truce in November and whose father Ohad is still held in Gaza, said she’s not convinced that the government will be able to close a hostage deal.

“We asked the prime minister to look us in the eyes and promise to do everything, and if it depends on him, not to give up until they return here alive. We received a nod and confirmation from him. We ask the Prime Minister to keep his commitment and bring them home. We understand that this is probably the last opportunity before we enter a large-scale war, and we want to see our loved ones at home,” she told reporters.

“Personally, I left with a heavy and difficult feeling that this isn’t going to happen soon, and I fear for my father’s life, for the girls who are there, and for everyone. With all the disinformation we hear, we no longer know what’s true and what’s not,” Ben Ami added.

“We had a long meeting with the prime minister,” said Yelena Trufanova, who was released from Hamas captivity on November 29 at the request of Russian President Vladimir Putin. “We shared all our pain, and I hope we found a listening ear. Mr. Benjamin Netanyahu looked me in the eyes and said he would do everything to bring my only son and all our loved ones home alive.”

She was kidnapped along with her 73-year-old mother Irena Tati and son Sasha from their home on Kibbutz Nir Oz on October 7. Her husband Vitaly was murdered during the rampage and Sasha is still being held hostage in Gaza.

It is now believed that 105 of the 251 hostages abducted by Hamas on October 7 remain in Gaza, including the bodies of 34 confirmed dead by the IDF. Hamas is also holding two Israeli civilians who entered the Strip in 2014 and 2015, as well as the bodies of two IDF soldiers who were killed in 2014.

Jacob Magid contributed to this report.