A US envoy has arrived in Israel to continue negotiations on a ceasefire with Lebanon’s Hezbollah movement, after reporting “additional progress” during a second day of talks in Beirut.

Amos Hochstein, who has led the Biden administration’s efforts to end more than one year of conflict, was expected to meet Israeli Prime Minister Benjamin Netanyahu on Thursday amid cautious optimism in Lebanon.

In what was seen as a positive development, he returned to Beirut on Tuesday after the Lebanese government and Hezbollah accepted a US-drafted proposal, although with some comments. He unexpectedly extended his visit by a day for further discussions.

After meeting Lebanese Parliament Speaker Nabih Berri, who has Hezbollah’s support to negotiate, Hochstein said he would go to Israel “to try to bring this to a close if we can”.

Details of the ceasefire proposal remained unclear, but some of the earlier differences included the formation of a supervision mechanism and Israel’s demand to be able to strike Lebanon if a deal was violated, which had been rejected by Lebanon.

Addressing foreign ambassadors in Jerusalem, Israeli Foreign Minister Gideon Saar said that, in any agreement, Israel “will have to maintain our freedom to act if there will be violations”, but he did not give details about what that would entail.

Since the conflict intensified in late September, Lebanese authorities have said any deal should be based on the United Nations Security Council Resolution 1701, which ended the 2006 war between Hezbollah and Israel.

The resolution includes the removal of Hezbollah’s fighters and weapons in areas between the Blue Line – the unofficial frontier between Lebanon and Israel – and the Litani river, about 30km (20 miles) from the boundary with Israel.

Israel says that was never fully respected, while Lebanon says Israel has violated the resolution by carrying out military flights over the country.

A potential deal would likely establish a timeline for the withdrawal of Israeli forces from the south of Lebanon and include the deployment there of thousands of additional troops of the Lebanese army.

But questions remain about how the Lebanese troops would operate against Hezbollah, which is heavily armed and backed by Iran, amid fears this could exacerbate sectarian tensions.

Israel’s stated goal in its war against Hezbollah is to allow the return of about 60,000 residents who have been displaced from communities in the country’s north because of the group’s attacks.

In Lebanon, the conflict has killed more than 3,500 people and forced more than one million from their homes, Lebanese authorities say. It has also caused an estimated $8.5bn (£6.7bn) in economic damages and losses, according to the World Bank.

Speaking in a pre-recorded televised speech, Hezbollah’s Secretary General Naim Qassem said the group had received the US proposal, clarified its reservations, and that it was allowing the talks to go ahead to see if they produced any results.

Qassem, whose speech was broadcast shortly after Hochstein made his comments, said the conditions for a deal were a complete cessation of hostilities and the preservation of Lebanon’s sovereignty, and that Hezbollah was ready for a long fight.

Israeli air strikes have destroyed large parts of Hezbollah’s infrastructure and killed many of its leaders, including long-time chief Hassan Nasrallah.

But after the initial shock, the group has reorganised itself. It continues to carry out daily attacks on Israel, although not with the same intensity, and confront invading Israeli soldiers in Lebanon’s south.

Amid the negotiations, Israel has stepped up its air campaign across Lebanon but, after three consecutive days of strikes on Beirut, there were no attacks on the capital during Hochstein’s visit.