Meetings aimed at preparing formal peace talks were to begin last week in Geneva but the Houthis refused to leave Sanaa to attend the consultations until a series of demands were met.
UN envoy Martin Griffiths told the Security Council he will be traveling to Muscat on Wednesday and later Sanaa and Riyadh to secure "a firm commitment from the parties to convene for continued consultations."
US supplied bomb that killed Yemeni children: report
The Houthis failed to turn up at the talks in Geneva over demands linked to the evacuation of wounded militants to Oman and safe passage for the return of the militant delegation to Sanaa.
The Yemen peace process will have "ups and downs", Griffiths told a council meeting, downplaying the setback as "temporary obstacles."
He asked the council to support his new shuttle diplomacy to "move back to the table with all speed."
It was the first UN bid to convene talks between the warring sides since 2016.
Yemen's Houthis say missile fired at Saudi Arabia's Najran region
The war in impoverished Yemen has killed 10,000 people and unleashed what the United Nations describes as the world's worst humanitarian crisis.
A Saudi-led coalition has been fighting Houthi militants in Yemen since 2015 to drive them out of the capital Sanaa and return President Abedrabbo Mansour Hadi to power.
The coalition has come under pressure to seek a political settlement from the United States, France and Britain which are increasingly ill at ease with the heavy toll that the bombing campaign has had on civilians.
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