Council members reaffirmed "their full support" for Griffiths' efforts.
The council is set to meet behind closed doors to discuss the assault launched on Wednesday on rebel-held Hodeida despite United Nations warnings of a looming catastrophe in Yemen, where millions are on the brink of starvation.
The ground offensive south of Hodeida is led by the United Arab Emirates, a key ally of both Saudi Arabia and the USA military.
The airport source said that Hadi returned to Aden just few hours after the arrival of Prime Minister Ahmed Obeid bin Daghr along with his ministers and other high-ranking officials in Aden's International Airport.
Late Thursday, Yemen's Foreign Minister Khaled Alyemany told a group of United Nations reporters in NY that military operations by the Saudi-led coalition are not taking place near the port but in an area close to the airport.
He also reported receiving a letter dated May 15 from Ukraine's United Nations ambassador indicating that its security service "prevented an attempt by two Iranian nationals to procure and transfer" to Iran components of a Kh-31 air-to-surface missile and related technical documents. "A full-scale assault on the port of Hodeida will tip the already dire humanitarian situation over the edge".
Huthi chief Abdelmalik al-Huthi urged his forces Thursday to keep fighting, saying they would recapture areas taken by pro-government forces and turn Yemen's western coastline into a "quagmire for the invaders".
Aid workers have warned the assault on Hodieda's port, known as the "mouth of Yemen", could shut down the vital route for some 70 percent of Yemen's food and humanitarian aid.
Hodeida is considered a lifeline for the country's war-ravaged population.
The Arab League said it supported the coalition, especially in Hodeidah.
The United Arab Emirates, a driving force in the coalition, said four of its troops were killed on the first day of the offensive.More news: Yemen's rebel-held Hudaida under assault from Saudi-led coalition
The ship was struck during the initial Saudi-led coalition assault on the Yemeni city of Hodeidah, which is spearheaded by the UAE and codenamed Golden Victory.
Foreign Ministry spokesman Bahram Ghasemi said Thursday that "such crimes will shatter the hopes for political endeavors and will add to the complexity of the situation".
Fighters backed by Saudi Arabia and the United Arab Emirates pushed toward the Yemeni port city of Hodeida on Thursday, on the second day of a battle that analysts say could be the bloodiest of the Yemen war. The resolution includes restrictions that took effect on January 16, 2016, on transfers to or from Iran of nuclear and ballistic missile material as well as arms.
According to a Reuters report, one USA intelligence official said that the UAE request for military and intelligence support was dismissed.
Therefore, he asserted, "it is vital that the port and other civilian infrastructure remain operational". But a successful operation would require swiftly capturing a city of 600,000 people, without inflicting damage that would destroy the port. "No direct attacks have been reported within Hodeida city itself, despite the overhead presence of fighter jets".
Aid groups warned a major humanitarian catastrophe was looming as fighting drew closer to Hodeida.
Ambassador Obaid Salem al-Zaabi made the comments during a news conference with journalists Thursday.
The officials declined to give death toll.
The Conflict Armament Research Center said earlier that the bombs are similar to those used by Iranian-backed Hezbollah in southern Lebanon and by insurgents in Iraq and Bahrain, suggesting an Iranian role in their manufacture.
The bulk of Yemen's food and aid enters through Hodeidah, raising fears that the military operation to retake the port could worsen the humanitarian crisis in the country, where more than 8 million people are facing starvation.
In response, the Yemeni forces dealt severe blows to the aggressors around the Hudaydah port.