According to ANI news reports, the first long-delayed withdrawal of Houthi forces began on Saturday from Hodeidah port, as well as another two smaller ports of Ras-Issa and Saleef.
The U.N. mission will monitor the redeployment, a first step towards concluding the peace agreement, the U.N. statement said, adding that it must be followed by "the committed, transparent and sustained actions of the parties to fully deliver on their obligations".
Lt. Gen. Michael Lollesgaard, the head of a United Nations mission monitoring the cease-fire in Hodeida, says Sunday that monitors will verify the rebels' withdrawal from the ports of Hodeida, Salif and Ras-Issa on Tuesday.
Yemen's Houthi rebels began pulling back from three ports on Saturday, the first steps on the ground since a ceasefire deal was agreed.
Sources close to the Iran-aligned Houthis told media that the ports were handed over to coast guard personnel who were in charge before the rebels took over nearly five years ago.
United Nations envoy "Martin Griffiths wants to achieve victory even if the Huthis hand over (the ports) to themselves", Taher said.
Some officials from the Saudi-backed, internationally recognized Yemeni government dismissed the pullout as a "show", but government negotiator Sadiq Dweid acknowledged it marked the start of implementing the Stockholm accord.
Yemen's Houthi movement on Saturday started withdrawing forces from Saleef port in Hodeidah under a United Nations -sponsored deal stalled for months, a Reuters witness said, reviving hopes for peace efforts to end the four-year war.
More agreed withdrawals were due to take place yesterday and today, opening up access to two more ports. The UN has said that a record 22.2 million Yemenis are in dire need of food, including 8.4 million threatened by severe hunger.
Under the redeployment pledge, Huthi militants said they also would move out of Yemen's main Red Sea port of Hodeidah by the end of the day on May 14.
The conflict in Yemen has already claimed tens of thousands of lives and put millions on the brink of starvation in what was already the poorest country on the Arabian Peninsula.
Hodeidah became the focus of the war past year when the coalition twice tried to seize its port to cut off the main supply line of the Houthis, whom they accuse of smuggling Iranian weapons, including missiles that have targeted Saudi cities. But they've yet to announce a withdrawal.
The port of Hodeida is one of Yemen's most important lifelines.
Both sides are expected to pull back further in a second phase.
The rebels seized Hodeidah and much of the country's north, including the capital Sanaa in late 2014, forcing President Hadi along with his government into exile in Riyadh.