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.
Yemeni officials say Saudi-led coalition airstrikes have hit most of the Houthi rebel defenses of Hodeida as part of the coalition's assault on the key Red Sea port town.
Yemen has been devastated by civil war, with the Houthi movement trying to take control of the country and forcing President Abdrabbuh Mansour Hadi to flee overseas.
Before dawn Wednesday, convoys of vehicles appeared to be heading toward the rebel-held city as heavy gunfire rang out.
Forces loyal to Yemen's exiled government and irregular fighters led by Emirati troops had neared Hodeida in recent days.
The U.S. has been supplying targeting information to the Saudi-led coalition, as well as refuelling their warplanes, but was not involved in military operations at the port, Pentagon spokesman Maj.
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Hodeida is home to 600,000 people and is the entry point for 70 percent of Yemen's imports, including vital aid supplies for civilians in the war-wracked country.
Hodeida is some 150 kilometers (90 miles) southwest of Sanaa, Yemen's capital, which has been in Houthi hands since September 2014.
The sources said Yemeni forces allied to the Saudi-led coalition - drawn from southern separatists, local units from the Red Sea coastal plain and a battalion led by a nephew of late former president Ali Abdullah Saleh - had advanced and were "at the doors" of Hodeidah airport.
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. They spoke on condition of anonymity as they weren't authorized to brief journalists.
Aid groups have expressed grave concerns for the safety of hundreds of thousands of civilians in the city whose lives could be at risk from the assault launched Wednesday.
Yemen's exiled government "has exhausted all peaceful and political means to remove the Houthi militia from the port of Hodeida", it said in a statement.
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The war pits the Houthis against the Western-backed Sunni Muslim states, which intervened in 2015 to restore the exiled government and thwart what Riyadh and Abu Dhabi see as expansionist aims Iran.
"Should we leave the Houthis smuggling missiles?"
State media in both Saudi Arabia and the United Arab Emirates did not immediately acknowledge the assault.
"I am extremely concerned about military escalation in #Hodeida & their humanitarian & political impact", he said in a tweet.
The UN on Monday withdrew all of its worldwide staff from Hodeida ahead of the impending assault, warning that any offensive would put millions of lives at risk. Already, Yemeni security officials said some were fleeing the fighting. Military sources said the deaths were caused by mines and snipers.
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