Yemen's humanitarian situation has worsened sharply, putting 8.4 million people on brink of starvation
UN-mediated Yemen peace talks hung in the balance as the government delegation warned that it would leave on Friday if representatives of the Houthi movement had not shown up.
The rebel delegation, still in Yemen's capital Sanaa, insisted the UN meets three demands before it travels to Switzerland.
They want the transportation of wounded rebels to Oman for medical care, the repatriation of rebels who have already received treatment there and a guarantee that the Houthi delegation will be allowed to return to Sanaa after the Geneva talks.
UN envoy Martin Griffiths, who earlier said the planned meeting offered a "flickering signal of hope" for an end to the years-long conflict, had to postpone the start of the talks and was left scrambling to save them.
"He continues to make efforts to overcome obstacles to allow the consultations to go forward," his office said in a statement, adding that Griffiths remained "hopeful" the rebels would come.
The Geneva talks are meant to be the first since 2016, when 108 days of negotiations between the government of President Abedrabbo Mansour Hadi and rebels failed to yield a deal.
A Saudi-led coalition intervened in Yemen's war against the Iranian-allied Houthis in 2015 to restore President Abd-Rabbu Mansour Hadi's internationally recognised government. Subsequent peace talks flopped.
Since then the humanitarian situation has worsened sharply, putting 8.4 million people on the brink of starvation and ruining the already weak economy.
UN Special Envoy Griffiths said in an earlier statement that he had discussed confidence-building measures with Yemen's foreign minister Khaled al-Yamani. The government delegation later met with Arab ambassadors who urged patience, a delegation source said.
"Today should have been the first day of our consultation. We are here, the Houthis are not accepting to come ... We think that they are committing a huge mistake by not engaging seriously in these peace talks," Yamani told reporters.
"We are not here indefinitely, we will not stay until the end of these consultations, we just wanted to prove to the Special Envoy that we are engaging. Tomorrow, in a few hours, we will take a decision on should we continue to stay in Geneva or should we withdraw," he added.
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Griffiths called for the Houthis to attend the talks, aimed at re-launching peace negotiations after nearly three years.
The world body wants the government and the Houthi movement to work towards a deal to end the war, remove foreign forces from Yemen, and establish a national unity government.
An inclusive political solution is required to end the conflict in Yemen, where people live in dire humanitarian, economic and security conditions, Griffiths said.
The Houthi movement's al-Masirah TV reported on Wednesday that the coalition had prevented the Houthi delegation from flying from Sanaa to Geneva. The Houthis have accused the UN of not keeping a promise to transport wounded on the flight.
On Thursday, a source from Sanaa airport said there was no plane there yet for the Houthi delegation.
Hamza al-Kamali, Yemeni deputy minister for youth, told reporters in Geneva earlier on Thursday that the flight clearance had been given three days ago.