The UN security council - which discussed the battle on Thursday - says the Hudaida port must remain open for humanitarian deliveries.
Ahmed al-Kawkabani, a Yemeni who leads a force known as the Tohama Brigade, told the Associated Press that his forces are now positioned in Dawar al-Hodeida, Arabic for "Hodeida roundabout".
The possibility of food running out is also among the residents' main fears. It was not clear what advances, if any, the ground troops backed by the Saudi-led coalition had made in Hodeida, or what the scope of casualties were.
The latest strike comes as Yemeni pro-government forces are locked in heavy fighting with rebels as they press a Saudi and UAE-backed offensive to retake the key aid hub of Hodeida.
However, the United Nations has not been able to determine when the missiles, components or related technology were transferred from Iran and if they violated UN restrictions, Guterres said in a biannual report on the implementation of UN sanctions on Iran.
The Red Sea port, controlled by the Iran-backed Houthi rebels, serves as the entry point for 70 per cent of the impoverished country's imports, but the coalition maintains that the rebels use it to smuggle weapons.
"They were united in their deep concern about the risks to the humanitarian situation and reiterated their call for the ports of Hudaida and Salif to be kept open", Nebenzia told reporters.
With logistical support from the USA, the Saudi-led coalition has been carrying out attacks inside Yemen since March 2015, killing at least 10,000 people, in an attempt to reinstate the internationally recognised government of President Abu-Rabbu Mansour Hadi. The Iranian-aligned fighters rifled through a military ledger from the vehicle before chanting their slogan: "Death to America, death to Israel, damn the Jews, victory to Islam!"
The United States rejected a request from the UAE for intelligence, mine-sweeping and airborne intelligence, surveillance and reconnaissance (ISR) assets for the Hodeidah operation, a UAE official said, speaking on condition of anonymity, while noting congressional opposition.
"We are not directly supporting the coalition offensive on the port of Hodeida", Rankine-Galloway said.More news: Beyoncé and JAY-Z drop surprise new album, Everything is Love: Stream
Mr Hadi fled the country that year after the Houthis tried to oust him from power, in part because he had neither a natural constituency in the deeply divided country nor loyalist fighters to protect him.
Saudi Arabia - Yemen's northern neighbor and Iran's principal rival in the region - soon intervened with a coalition that includes the United Arab Emirates and gets logistical support from the USA, seeking to reinstate Yemen's government. At least 10 UAE ships carrying 13,500 tons of food and aid, as well as three flights, were planned for Hodeida, it said.
"They did not encourage us, but did not discourage us", the official said.
"The situation is dire and we don't know how it will end", said Khadija, a teacher in Hodeidah.
The Security Council, the only United Nations body with the power to pass legally binding resolutions, on Thursday brushed aside a call by non-permanent member Sweden for a pause in fighting to allow for talks on a rebel withdrawal from Hodeida.
On Wednesday, the Saudi-led coalition launched a major operation to recapture the port of Al-Hudaydah - a vital supply line for humanitarian aid entering the war-torn country.
Almost 40 people have been killed in fighting outside Yemen's Hudaydah port, medical sources say.
The kingdom, which leads a military coalition that has fought the insurgents since 2015, usually claims to intercept most missiles.
Emirati Minister of State for Foreign Affairs Anwar Gargash said that the battle over Hodeida is essential to break a stalemate in the war, which otherwise could drag on for years.
The ambassador's comments contradict the conclusions of a United Nations panel of experts that said it was unlikely the Houthis were using the port for smuggling arms.