Much work remains, admitted John Kirby, spokesperson for the US National Security Council.
Talks in the Qatari capital, where among others CIA chief William Burns is participating, are expected to continue on Friday.
Hamas representative Osama Hamdan announces that the Islamist movement did not participate in Thursday's meetings, but is ready to join if Israel shows willingness to negotiate.
Hamas has demanded that the peace plan presented by US President Joe Biden earlier this spring be implemented.
If the mediators succeed in forcing the occupying power (Israel) to say yes, then we will do so too, but so far there are no signals about that, says Hamdan to AFP news agency.
Demanding withdrawal
Later, Hamas representative Hossam Badran repeated the demand that a ceasefire agreement must include a "full withdrawal" of Israeli forces from Gaza.
Exactly who is participating in the talks in Doha is not confirmed, but it has been previously stated that both the US and Israel will be represented at a high level.
The New York Times' sources close to the US leadership say that Israel should now be more ready for a ceasefire, since the country's military has achieved so much in the Gaza Strip.
Israel has been able to disrupt Hamas, kill a number of its leaders and greatly reduce the threat that existed before (the major terrorist attack) on October 7, says Joseph Votel, former chief of the US regional command Centcom, to the newspaper.
Israel cannot hope for much more.
Hamas is drained of power, but not eradicated. The Israelis may never be able to achieve total elimination of Hamas, says Ralph Goff, former high-ranking official in the Middle Eastern intelligence service.
33 hostages
One question is, however, how far Israel's emissaries can and will go.
It's about, for example, what Israel can agree to regarding hostages. According to an earlier draft, the terrorist-listed Palestinian Hamas is to release 33 people from captivity in a first phase.
But in the spring, it was not specified whether these 33 must be alive, or if it is enough for Hamas to hand over "human remains". Now, Israel is said to demand that they are alive, and also want some soldiers to be among them.
The stakes are high. The entire Middle East has been bracing for Iran's revenge since Hamas leader Ismail Haniya was killed, likely by Israel, in Iran's capital Tehran. But according to US President Joe Biden, Iran may refrain from the feared revenge attack if a Gaza agreement is reached.
