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

Freed Israeli hostage Aviva Siegel, whose husband Keith is still held captive, said she was feeling "happier and more optimistic than before" after meeting with U.S. Secretary of State Antony Blinken on Tuesday. After hearing from Blinken, Siegel said she wants to believe that her husband, an American-Israeli citizen, will be home soon.

Siegel said Blinken gave her the impression that Israeli Prime Minister Benjamin Netanyahu wants the deal that's on the table, but cautioned that it sounds like hard work needs to be done to close the gaps with Hamas.

Outside of the building where families of the hostages met Blinken, crowds chanted "SOS USA" and waved American flags, with a message to Blinken that the U.S. government is their main lifeline to bring the hostages home.

Despite the optimism from the Biden administration, momentum appears to have slowed. Talks in Doha did not end with any major breakthrough, and a U.S.-presented "bridging proposal" has been dismissed by Hamas, who accuse Netanyahu of adding new conditions and obstacles, including the rejection of a permanent ceasefire.

Blinken travels to Egypt tomorrow, where he says he'll get the latest from Egyptian and Qatari leaders on Hamas' intentions.

-ABC News' Britt Clennett, Guy Davies, Anna Burd and Tomer Slutzky

At least eight people have been injured from Israeli strikes on the Beqaa Valley, the Lebanese Ministry of Public Health said Monday.

Among the wounded are six Lebanese citizens, a 5-year-old Syrian girl and a 15-year-old Syrian girl. All of them were treated in the emergency room at nearby hospitals, the Ministry of Public Health said.

There were seven Israeli strikes in southern Lebanon on Monday, according to Lebanese local media.

In two separate statements Monday, the IDF acknowledged conducting operations "in the area of Houla in southern Lebanon," "in the area of Beqaa in Lebanon" and in “the area of Tayibe in southern Lebanon.”

-ABC News’ Ghazi Balkiz

Secretary of State Antony Blinken said Monday afternoon that Prime Minister Benjamin Netanyahu confirmed in their earlier meeting that he supports and would accept the U.S.-backed bridging proposal that emerged after meetings in the region last week, and that Israel would send negotiators.

"It's now incumbent on Hamas to do the same," Blinken told reporters on his ninth trip to the region since Oct. 7. "And then, the parties, with the help of the mediators -- the United States, Egypt and Qatar -- have to come together and complete the process of reaching clear understandings about how they'll implement the commitments that they've made under this agreement. But the next important step is for Hamas to say yes.”

Blinken acknowledged the "complex issues" would still "require hard decisions by the leaders," but said he still felt "a real sense of urgency, here, across the region, on the need to get this over the finish line and to do it as soon as possible."

But given Hamas' unwillingness to engage in the latest round of negotiations, the secretary was pressed on whether there was any real hope the militant group would sign on to the proposal.

"Tomorrow, when I see the leaders of both Egypt and Qatar, I'll get the latest from them on what they are hearing," he responded. "I can't speculate on exactly what Hamas' intentions are. We've seen public statements, but we've seen public statements before that don't fully reflect where Hamas is."

Blinken will then travel to Qatar after visiting Egypt on Tuesday, he said.

-ABC News' Shannon K. Kingston

Hamas’ military wing, the Al-Qassam Brigades, took responsibility for an attack in Tel Aviv on Sunday evening that left one person injured and the bomber dead.

In a joint statement released Monday, Israeli Police and Shin Bet said the explosion had been a terror attack.

"It can now be confirmed that this was a terror attack involving the explosion of a powerful explosive," the Israeli police and Shin Bet said. "As a result of the explosion, a passerby was moderately injured (according to medical sources) and was transported to receive medical treatment."