The Saudi-backed government and the Houthi rebels have shown a "renewed commitment" to work on a political solution and have given "firm assurances" that they will attend the talks, Martin Griffiths told the Security Council.
"With this in mind, I intend to reconvene the parties shortly and to do so in Sweden," he said. "I believe we are close to resolving issues to make this happen."
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The Saudi-led coalition has agreed to "logistical arrangements" to pave the way for talks including medical evacuations out of rebel-held Sanaa, he added.
Griffiths announced he was close to reaching a deal on an exchange of prisoners and detainees, in a further sign that confidence-building measures were being taken ahead of planned talks.
"This is a crucial moment for Yemen," he said, warning that a flareup of fighting on the ground could derail the peace effort. No date for the talks was announced.
The United Nations had announced talks in Geneva in September that never materialised after the Houthis put forward last-minute demands.
Griffiths plans to travel to the rebel-held capital of Sanaa next week to finalize arrangements and offered to travel with the Houthi delegation to Sweden "if that's what is needed."
The United States and other western powers have called for a ceasefire and talks on ending the war, which has unleashed the world's worst humanitarian crisis.
Eight million people are affected by severe food shortages, according to UN officials, who warn that up to 14 million - or half of Yemen's population - are at risk of famine.
Back from a visit to Yemen, the head of the UN World Food Programme warned Friday that Yemen faces a full-blown famine in about six months because of the economic collapse from the war.
"This is not on the brink of catastrophe, this is a catastrophe," David Beasley told reporters.
The Saudi-led coalition has been waging a war in Yemen since March 2015 to push back the Iran-backed Houthis and restore to power Yemeni President Abedrabbo Mansour Hadi, whose government is recognised by the United Nations.
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As pressure mounted on the warring sides, Yemeni pro-government forces have suspended an offensive on Hodeida, a major entry point for humanitarian aid and vital goods to Yemen.
The UN envoy said he will travel to Hodeida next week to discuss plans for the United Nations to take over control of the port and oversee the arrival of aid deliveries and supplies.
This would address concerns from the coalition that the weapons are being smuggled into Yemen for the Houthis through the port.
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