Griffiths was in Amman following an intense round of shuttle diplomacy this week with Yemen's Huthi rebels, who control the port, Saudi Arabia and the United Arab Emirates, whose forces are backing the Hodeida offensive.
Earlier Tuesday, Yemeni officials said the United Nations had pulled its worldwide staff out of Hodeida, a crucial Red Sea port city now controlled by the Houthis.
Saudi-owned satellite news channels have announced the start of the Saudi-led coalition's assault on Hodeida. The U.N. says some 600,000 people live in and around Hodeida, and "as many as 250,000 people may lose everything- even their lives".
More than 10,000 people have been killed in Yemen's civil war, which has displaced 2 million others and helped spawn a cholera epidemic.
Before dawn, convoys of vehicles appeared to be heading toward the rebel-held city on the Red Sea, according to videos posted on social media.
United Nations spokesman Stephane Dujarric told reporters at United Nations headquarters in NY on Wednesday that aid partners are delivering 70,000 rapid response kits to "humanitarian service points" in and around Hodeida for newly displaced families.
Troops in a Saudi-led coalition have captured a town south of the Yemen port city of Hodeida as fierce fighting and air strikes pounded the area, officials said.
Hodeida is some 150 kilometers (90 miles) southwest of Sanaa, Yemen's capital, which has been in Houthi hands since September 2014.
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They also are a means of showing allied solidarity, which is part of the psychology of deterring enemy attack. He said to the North Korean leader in televised footage: "Come to Russian Federation ".
The port remained open on Thursday, according to the Norwegian Refugee Council, which cited information from the U.N. There were "four vessels filled with food and fuel at berth" and another five vessels anchored nearby, it said.
"I think the Emiratis have done a good job in presenting compelling arguments about why an operation (on Hodeidah) could in the end tip the balance and apply enough pressure to bring the Houthis to the table", a Western diplomat said on Monday.
Yemeni residents in the Red Sea port city of Hodeida say the Saudi-led coalition has dropped leaflets advising them to stay away from military and security points, and to stay in their homes, amid the coalition's assault. They spoke on condition of anonymity as they weren't authorized to brief journalists.
Minister of State for Foreign Affairs for the United Arab Emirates, Anwar Gargash, speaks at an event at Chatham House in London, Britain July 17, 2017.
The aid group Oxfam said humanitarian organizations received warnings over the weekend for staff to evacuate Hodeida by Tuesday ahead of an offensive. The sound of heavy, sustained gunfire clearly could be heard in the background. We are concerned about missiles and shells.
Griffiths warned in a statement earlier Wednesday that "further military escalation will have serious consequences on the dire humanitarian situation in the country" and hinder his efforts to restart political negotiations.
"We do not provide any additional support to the Saudi coalition's military operations", he said.
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