Israeli negotiators have expressed "cautious optimism" regarding the talks in Doha, Qatar to secure the release of 115 captives still being held by Hamas in the Gaza Strip.

The team updated Israeli Prime Minister Benjamin Netanyahu on Saturday regarding the discussion that took place over the previous two days, according to a statement by the Prime Minister's Office.

"The team expressed cautious optimism to the prime minister regarding the possibility of progress on the deal, in accordance with the updated American proposal (based on the May 27th framework), including components acceptable to Israel," said the PMO.

"It is hoped that the heavy pressure on Hamas by the United States and the mediators will remove its opposition to the American proposal and will lead to a breakthrough in the talks," added the statement.

There is a consensus among the moderating countries—the United States, Egypt and Qatar—that the diplomacy is “in the endgame,” a senior American official told reporters on Friday from Doha.

The official called the latest negotiation session “the most constructive in 48 hours we’ve had in many months,” echoing a joint statement that the three countries released earlier in the day.

The “intensive” negotiations were “serious and constructive and were conducted in a positive atmosphere,” Washington, Cairo and Doha said on Friday.

In the coming days, the parties involved will work on “details of implementation, including arrangements to implement the agreement’s extensive humanitarian provisions, as well as specifics relating to hostages and detainees,” the three countries added, referencing Palestinians detained in Israel on terror charges.

Israel's Channel 12 reported on Saturday that the American proposal does not call for an ongoing Israel Defense Forces presence along the Egypt-Gaza border or a mechanism to prevent the return of Hamas terrorists to northern Gaza, both of which have reportedly been demanded by Netanyahu.

The senior U.S. official noted on Friday that "there’s still more work to do," adding: "This is a very difficult, complex deal. It is wrenching because this is about getting hostages out of Gaza. There are elements of the deal that are uncomfortable. Just like any deal like this, it is far from perfect.”

Another summit is reportedly scheduled for Cairo on Thursday, with U.S. officials saying the goal is to finalise a deal by the end of the week.