A military official told AFP on Friday the government alliance had approached to within just two kilometres of Hodeida's global airport, just inland from the port.
Depriving the Houthis of their control of Hodeida port, at the Yemeni government's request, means that the Houthis will no longer be able to impose their will at the barrel of a gun.
Streets elsewhere in the city were empty despite the Eid holiday marking the end of the Ramadan fast. "We live days of terror that we have never known before".
While the Yemeni army and popular committees are fighting off the assailants and countering attacks on residential areas, global humanitarian organizations have warned of dire consequences of any escalation of violence around Hudaydah.
Houthi forces had entered homes overlooking the main road to go onto the roofs.
The government side sent significant reinforcements to the front on Thursday, according to military sources and the AFP correspondent.
The attack was launched on the coastal road that leads up from the government-held towns of Khokha and Mokha.
Coalition forces were now just 2 km from the airport, Emirati Ambassador to the United Nations Obeid Salem Al Zaabi told reporters in Geneva. The Houthis are backed by Iran.
Sweden, a non-permanent council member and leading voice on humanitarian issues, called on the council to demand an immediate halt to the assault to allow time for talks on a rebel withdrawal from Hodeida Port.
The battle has since killed at least 139 fighters, according to medical and military sources. 30 rebels were said to be killed.
Ms Nusseibeh said the offensive to oust the Houthi rebels was "a critical step toward achieving a political solution to this conflict because we know there is no military solution".
The senior UAE official said the coalition gave Griffiths a chance to negotiate with the Houthis and get them to abandon the port but that the Houthis rejected those efforts. Worldwide aid groups have long warned against an offensive on Hodeida because the port serves as the entry point for 70 percent of Yemen's imports, as the country teeters on the brink of starvation.
On Thursday, authorities said the Red Sea lifeline remained open to shipping. "If you cut that port off, we have a catastrophe on our hands".
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Despite the joy in the faces of most of the population in Sana'a, citizens fear a consequent disruption of the port's work and the disruption of the flow of goods and fuel to Sanaa and other provinces under militia rule. "And we have five other ships standing by waiting outside to enter", port director Dawood Fadel told AFP.
"We are not planning to destroy the infrastructure", he added. "We can hear the fighting coming close and the situation is really changing for the worse". "The sea port is totally out of operations today", he told reporters on Thursday.
"The Houthis have gathered a large number of their fighters in the city and on the outskirts", Bassim al-Jenani, a freelance Yemeni journalist, told Middle East Eye.
A report by the Saudi-owned satellite network Al Arabiya has quoted commander Abu Zarah al-Mahrami as saying government forces are "within meters" of the airport.
The rights group called on all parties of the conflict to minimize civilian harm during the fight for Hodeida. The worldwide aid group warned of a "high risk of a second outbreak" should water supplies be disrupted.
The United Arab Emirates says it has started flying and sailing aid to Yemen's port city of Hodeida, where its troops in a Saudi-led coalition are fighting Shiite rebels.
Council members "reiterated their call for the ports of Hodeida and Saleef to be kept open", said Nebenzia.
United Nations special envoy Martin Griffiths, meanwhile arrived in Sanaa in an effort to broker a cease-fire.
UAE Minister of State for Foreign Affairs Anwar Gargash has called on the global community to pressure the Houthis to abandon the port.
The coalition pounded Houthi positions in Yemen's Hodeidah for a second day on Thursday during a military assault aimed at seizing the main port to prevent the Houthis from bringing in missiles from Iran.
The United States has turned down three requests within the past 24 hours from the UAE to provide more support, including minesweepers.
Dozens of combatants have been killed since the offensive began on Wednesday and the United Nations has voiced concern for the vital aid shipments that pass through the city's docks.