The destruction of the port of Hodeida, a vital lifeline for millions of starving civilians in Yemen, would "create a catastrophic situation", Antonio Guterres told French radio station France Inter.
In a Friday statement, Yemen's internationally recognized government based in the southern city of Aden said its forces are advancing toward the north and west of Hodeida and across all fronts with cover from the coalition.
The Saudi-backed government forces vowed to continue to push forward in an attempt to seize full control over the city's strategic port, but Houthis face the offensive with stiff resistance.
Almost three quarters of Hodeida's estimated 600,000 people have escaped since June, a figure that underscores the dire situation in the city that serves as the main entry point for food and aid, the UNHCR said.
The developments came after the offensive on Hodeida by pro-government forces and the Saudi-led coalition, including the UAE, appeared to stall.
The UK and the USA are major suppliers of arms to Saudi Arabia, which leads a military coalition backing the Yemen government in its fight against Houthi rebels.
The Saudi-led coalition has also come global pressure to end the conflict following the killing of Saudi dissident journalist Jamal Khashoggi in his country s consulate in Istanbul on October 2.
Residents reported Tuesday that fighting in the city had slowed, and rebel media - which regularly claims attacks on loyalists - did not report any new clashes in Hodeida itself.
The United States, which for years provided military training and aerial refuelling for the Saudi-led coalition, on Saturday announced it would end its in-flight refuelling support for the alliance.More news: Asia Bibi: Boris Johnson among MPs urging government to give asylum
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The lull coincided with a visit by British Foreign Minister Jeremy Hunt to Saudi Arabia, to press for an end to the almost four-year war which has killed more than 10,000 people.
"There is only one entrance left into the port, and that is the main gate that leads to Mina Street that trucks use", one employee, who requested anonymity, told AFP.
"We are not hearing explosions like we have for the past two weeks".
Other residents have said they feared being trapped in the city, where only one major exit route is still open to traffic and the transport of food and aid, on the northern edge of the city.
Hodeida port is under a near-total blockade by Saudi Arabia and its allies, who accuse Iran of smuggling arms to the Huthis.
"We welcome early convening of UN-led talks in Sweden", the UAE state minister for foreign affairs, Anwar Gargash, tweeted in English.
Britain is seeking support among regional partners for new action at the UN Security Council for peace talks in Yemen.
The United States, Britain and France - three of the main arms suppliers to Saudi Arabia - have also called for an end to almost four years of conflict, particularly in the Red Sea city of Hodeida.