More than 25,000 people have been displaced in the recent fighting, which has been the fiercest in Yemen's key port since a UN-brokered truce in December 2018.
Mohammed Ali al Houthi, the head of the rebels' Supreme Revolutionary Committee, says the pullout started on Saturday at 0700 GMT. But the Yemen government says the rebels are faking withdrawal.
The UN's current peace push in Yemen is the biggest since 2016 when 108 days of negotiations in Kuwait failed to yield a deal between the government and Iran-aligned Houthi rebels.
Yemen's economy and population of 22 million people depend almost entirely on imports which could face disruption as fighting rages on around the city of Hudaida, home to country's main port.
The UAE says the pause in hostilities is aimed at pursuing negotiations for an unconditional Houthi withdrawal from Hudaida but warns that full military action could resume.
Both Yemen's Shia rebels, known as Houthis, and the country's Saudi-backed government confirmed their willingness to come to the table to restart negotiations with the UN Security Council.
Pro-government forces captured the Al Faza district while advancing towards the nearby Al Tahita Directorate. An earlier meeting between the UN's special envoy and Yemen's exiled president failed to bring about any ceasefire.
Yemen’s internationally-recognized government and its allies accuse the Houthis of using Al-Hudaida to smuggle weapons into the country from Iran.
Residents said coalition aircraft were bombing Houthi positions on roads leading to the airport as the group dug in against an onslaught by the Arab alliance to take the city, the Houthis' main port and the lifeline for millions of Yemenis.
The assault on the airport of Yemen's main port city has been underway for the past four days as the Saudi-led coalition attempts to gain control of Hudaida, which is the main conduit for aid.
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