The spokesman of the Arab coalition Col Turki Al Malki stressed in a press conference on Monday the need to protect civilians and sustain the flow of aid.
The UN and aid agencies have pleaded with the Saudi-led force against an attack on Hodeida, warning that it could lead to mass civilian casualties and starvation.
"We are doing everything we can through diplomatic channels to discourage an assault on Hodeidah".
The Western diplomat said there were "a couple of last ditch attempts" to try to see whether there were any options to de-escalate, including a plan to "see some sort of a temporary ceasefire", but that would involve the Houthis putting their weapons down or potentially even moving out of the port.
Following the closed-door meeting, Russian Ambassador Vassily Nebenzia, who is council president this month, called for de-escalation and said the top United Nations body would be "closely" following developments.
United Nations aid chief Mark Lowcock said an attack on Hodeida would be "catastrophic" and that aid agencies were hoping to "stay and deliver" in Yemen, which the United Nations describes as the world's worst humanitarian crisis.
"We are, at the present moment, in intense consultation", the UN's secretary general, António Guterres, told reporters.
Yemen relies on imports for 90 percent of its food and 70% of those transit through Hodeida, Lowcock said.
U.S. Defense Secretary Jim Mattis, speaking to reporters at the Pentagon, acknowledged the "humanitarian crisis" in Yemen and said he had met twice with Griffiths and also spoken to him by phone.
A picture taken during a tour organised by Yemeni loyalist forces backed by Saudi and Emirati forces on June 3, 2018, shows what they said were detained Huthi fighters in a coast guard base near the port of Mocha which is under the control of pro-government forces.
The United Nations pulled all of its global staff out of Al Hodeidah early on Monday morning.
Mark Lowcock, the head of the UN's office for the Coordination of Humanitarian Affairs, said the consequences of the suspension of operations at the port of Hudeida would be "catastrophic".
More than 10,000 Yemenis have been killed since the war began, tens of thousands have been wounded, and another two million people have been displaced.
The Saudi-led coalition of mostly Persian Gulf nations has been carrying out airstrikes against the Houthis at Hadi's request since March 2015.
The conflict has left almost 10,000 people dead in Yemen, already the Arab world's poorest country.