As Israeli and Hamas officials continued indirect talks on Monday for a cease-fire and hostage release deal, Hamas representatives indicated that the group had approved an Israeli list of 34 hostages to be released in the first stage of an agreement, conveying a sense of momentum.

But the Israeli government quickly dampened any notion that Hamas’s acceptance of the list constituted a significant breakthrough toward ending the war in Gaza.

Here’s what we know about the list of hostages and the state of the talks, after months of false starts and failed rounds.

What Hamas is saying

A Hamas representative briefed on the negotiations told The New York Times on Monday that if a deal is reached, Hamas agrees to release the 34 hostages whose names appear on a list that Israel provided via mediators. The representative spoke on the condition of anonymity because the talks are being held in secrecy.

The United States has long been pushing for a deal and is trying to facilitate one with the help of Qatar and Egypt, the two main countries mediating between Israel and Hamas.

The Reuters news agency reported on Sunday night that Hamas had approved the list presented by Israel. The report cited an unnamed Hamas official who added that any hostage deal was contingent upon Israel agreeing to a timetable for withdrawal from Gaza and a permanent cease-fire, in line with Hamas’s longstanding demands.