Israel-Gaza live updates: Sirens sound in Tel Aviv as Hamas fires rocket from Gaza

Before wrapping up his trip to the Middle East, Secretary of State Antony Blinken spoke to members of the media on a tarmac in Qatar, emphasizing the urgency behind the latest push for a hostage release and cease-fire agreement in Gaza.

"Our message is simple, it's clear, and it's urgent. We need to get the cease-fire and hostage agreement over the finish line, and we need to do it now," he said.

"Time is of the essence, because with every passing day, the well-being and lives of the hostages are in jeopardy. Time is of the essence, because every single day, women, children, men in Gaza are suffering," he said. "And time is of the essence because with every passing day, there's the danger of escalation in the region."

Blinken said Qatar and Egypt remain "in direct contact with Hamas working to bring it along" with the bridging proposal, but he again made clear that even if that were to happen, all sides would still need to complete "detailed implementation agreements that go along with putting the cease-fire into effect."

Israeli Prime Minister Benjamin Netanyahu told representatives of the hostage families that "Israel will not under any circumstances withdraw from the Philadelphie axis and the Netzer Corridor despite tremendous pressure to do so," according to a release from the Tikva Forum of hostage families.

"I'm not sure there will be a deal, but if there is a deal -- the deal will be one that preserves the interests that I repeat over and over again, which are the preservation of Israel's strategic assets. The demand that there be negotiations, even in the future -- a demand that is unacceptable, even if there is a deal -- after 42 days, we will return to fighting immediately, until the elimination of Hamas, even while negotiating the next steps," Netanyahu told hostage families.

"The military pressure will continue with full force, this is the only thing that makes Hamas drop its unreasonable demands," Netanyahu added.

Asked by ABC News, Netanyahu's office said they had nothing to add beyond a press release that did not include the comments.

-ABC News' Jordana Miller and Dana Savir

The bodies of six hostages recovered by Israeli Defense Forces were found in an area where Israeli forces had previously operated in Khan Younis, IDF spokesman Daniel Hagari said in remarks to the press Tuesday.

The circumstances of the hostages' deaths are being investigated, Hagari added.

"The operation was led by the 98th Division, which returned to operate in the Khan Younis area for the third time a few weeks ago, this time with new intelligence and new targets," Hagari said.

-ABC News’ Jordana Miller

As U.S. Secretary of State Antony Blinken continues his meetings with Middle Eastern mediators working toward a Gaza cease-fire deal, a senior U.S. official traveling with his delegation is criticizing comments about the process emanating from both Israel and Hamas, casting them as unproductive.

The official, who spoke on the condition of anonymity, addressed reports that Israeli Prime Minister Benjamin Netanyahu took a hawkish stance in a meeting with hostage families, telling them he was "unsure" there would be an agreement to return those held captive in Gaza and that he would not withdraw troops from strategic areas of the enclave -- a key sticking point in negotiations.

"Maximalist statements like this are not constructive to getting a cease-fire deal across the finish lines," the official said. "And they certainly risk the ability of implementing and working level and technical talks to be able to move forward when both parties agree to a bridging proposal."

The senior official also dismissed Hamas’ latest statement accusing the U.S. of "blind bias towards Israel."

"They say a lot of things and they do a lot of different things," the official said, suggesting the militant group’s ongoing engagement was an indication that Hamas was still involved in the negotiation process despite its insistences to the contrary.

-ABC News' Shannon K. Kingston