The UN statement comes after Houthis took control of a large area south of Hudaida, a key port where the warring sides agreed on a ceasefire in 2018.
More than 6,000 people have been displaced due to a recent Houthi rebel advance near Yemen's lifeline port of Hudaida, the United Nations said.
Almost 5,000 people were displaced to Khokha, over 100 kilometres south of Hudaida, while about 1,300 people were displaced further south to the Red Sea coastal town of Mokha, the UN Office for the Coordination of Humanitarian Affairs said on Sunday, citing Yemeni government sources.
"No displacement has been reported within the areas that came under control of the de facto authorities," it said in a statement, referring to the Houthis.
Citing aid partners on the ground, it said a 300-tent site for displaced people had been set up in the Khokha district, while the authorities were reportedly looking for another site to cope with the influx.
But the UN also said the Houthi advance could result in "improved movement for civilians" between the provinces of Hudaida and Sanaa, and along roads connecting Hudaida city with other districts.
Two military officials said that fighting also erupted on Saturday when the rebels tried to push farther south into government-controlled territory, but loyalist forces repelled the advance.
Clashes continue despite 2018 ceasefire
The Houthis on Friday took control of a large area south of Hudaida, a key port where the warring sides agreed on a ceasefire in 2018, after loyalist forces withdrew.
The Hudaida ceasefire was agreed on at Yemen's last peace talks in Sweden in 2018, but clashes have since broken out between the rebels and pro-government troops around the city.
A Saudi-led coalition intervened in Yemen in 2015 to shore up the government, a year after the Houthis seized the capital Sanaa.
Tens of thousands of people, mostly civilians, have been killed and millions displaced, in what the UN calls the world's worst humanitarian crisis.
The Iran-backed group has also been engaged in a sustained attempt to capture Marib, the government's last stronghold in the north.
The UN last week called on "all parties to the conflict to ensure the safety and security of civilians in and around those areas where shifts in frontlines have taken place".