Residents said battles were raging on the coastal road leading to the densely populated city centre from the airport, with Apache helicopters of the Western-backed Saudi-led coalition providing air support.
The Saudi-led coalition fully captured Hudaida airport from Iran-backed Houthi rebels, Yemen’s government military and the coalition said on Tuesday, in a significant step to a takeover of the key port city.
Saudi-owned Al Arabia network reported that residents of the strategic city said battles were also raging on the coastal road leading to the densely populated city centre from the airport, with Apache helicopters of the Western-backed coalition providing air support.
"We can hear the sounds of artillery, mortars and sporadic machine gun fire. The Houthis have been using tanks," a resident close to the coastal strip told Reuters by telephone, asking not to be identified.
"Water has been cut off to many of the areas near the Corniche area because the Houthis have dug trenches and closed water pipes. Many people are fleeing these neighbourhoods and going deeper into the city centre."
The Houthis said they fired a missile targeting a facility of oil giant Saudi Aramco in Asir in southwestern Saudi Arabia, Houthi-run media reported.
An Aramco official said the company's facilities were safe and operating normally.
Saudi and UAE state media accused the Houthis of shelling civilian districts. Residents said the Houthi tanks were targeting coalition forces.
The Arab alliance launched the onslaught on Hudaida, the Houthis' sole port, on June 12 to try and turn the tables in a long-stalemated proxy war between Saudi Arabia and Iran that has exacerbated turmoil across the Middle East.
"They have stormed the airport," an anti-Houthi Yemeni military source said on Tuesday.
The escalation in fighting has wounded and displaced dozens of civilians and hampered humanitarian agencies trying to send vital aid to million of Yemenis via the Red Sea port.
Tuesday's battles spread panic among local inhabitants.
"My children are terrified. The fighting and the sounds of explosions are everywhere and we are stuck in our house in the district of Rabsa with no running water," Iman, a 37-year-old mother of two, said tearfully.
"What have we done for all of this?”
Mohamed Sharaf, 44, a civil servant, said he had sent his entire family to Sanaa, the Houthi-held inland capital, several days ago and he was getting ready to leave himself. "There is death and destruction everywhere in this city.”
The United Nations fears the offensive will worsen what is already the world's most urgent humanitarian crisis, with 22 million Yemenis dependent on aid, and an estimated 8.4 million believed to be on the verge of starvation.
UN officials estimate that 600,000 people live in and around Hudaida and that in a worst-case scenario the battles could cost up to 250,000 lives. A senior UN humanitarian official told Reuters a leading determinant in arriving at that estimate was the risk of a cholera outbreak.
The war has killed more than 10,000 people in all.
Hudaida port remained open on Tuesday with the UN World Food Programme hastening to unload three ships containing enough food for six million people for one month, WFP spokeswoman Bettina Luescher told reporters in Geneva.
The Arab states said their aim was to seize the airport and port quickly and avoid street battles in the city centre. But the Houthis were well dug into Hudaida as it constitutes the key supply line to territory they control including Sanaa.
UAE Minister of State for Foreign Affairs Anwar Gargash said, "We are waiting for them (Houthis) to realise the sort of military and psychological blow that they got with the airport ... we are giving them some time to come and decide if they want to save the city ... and pull out."
Diplomacy aimed at reducing the bloodshed appeared stalled.
Martin Griffiths, UN peace envoy for Yemen, left Sanaa on Tuesday after talks with Houthis. He made no comment.
But Senior Houthi official Dhaifullah Shami said on Twitter that Griffiths had "failed to achieve his goal and left Sanaa disappointed, without obtaining concessions."
Another Houthi official, Mohammed al Bukhaiti, denied an assertion by Gargash on Monday the talks with Griffiths were about handing over Hudaida, calling this idea "unrealistic."
Bukhaiti told Reuters that Griffiths instead had discussed "a comprehensive political solution that addresses ... all fronts and not only Hudaida."
Stephane Dujarric, spokesman for the UN Secretary-General, told reporters that Griffiths was going on to Jeddah in Saudi Arabia to meet coalition and Yemeni government officials.
The coalition intervened in Yemen's war in 2015 to try and unseat the Houthis, restore the internationally recognised Yemeni government in exile and thwart what Riyadh and Abu Dhabi see as Iran's expansionism in the region.
The Houthis, who control the most populated regions in the chronically unstable nation of 30 million people, deny they are puppets of Iran. They say their movement reflects a popular revolt against state corruption and foreign meddling.