Saudi and Emirati-backed forces have begun an assault on the Houthi rebel-held Yemeni port city of Hudaydah.
Over 10,000 people have been killed in Yemen's civil war.
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.
Meanwhile, diplomats in the region told The New York Times that "only pressure from Washington, which sells tens of millions of dollars of weapons to the United Arab Emirates and to Saudi Arabia every year, could stop the assault".
On Thursday, a senior UAE official said the Trump administration has rejected requests for military assistance in the coalition attack on Hodeida.
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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.
Some 70% of Yemen's food enters through the port, as well as the bulk of humanitarian aid and fuel supplies.
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.
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.
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.
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Emirati-led troops have advanced along the southwestern coast to the outskirts of Hodeidah under a coalition strategy to box in the Houthis in the capital Sanaa and choke off their supply lines to force them to the negotiating table. He said the U.N. Security Council must act to secure a cease-fire before the people of Hodeida "suffer the same fate as those in Aleppo, Mosul or Raqqa".
The attack is expected to lead to a protracted and bloody confrontation, putting "literally hundreds of thousands of people... into a life-threatening situation", Lise Grande, United Nations humanitarian coordinator for Yemen, told USA radio station NPR.
The Red Sea port serves as the entry point for 70 percent of Yemen's imports as the country teeters on the brink of starvation.
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.
The United Nations and other aid groups had pulled their global staff from Hodeida ahead of the rumored assault. The sound of heavy, sustained gunfire clearly could be heard.
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"We do not provide any additional support to the Saudi coalition's military operations", he said.