A shutdown of the port could send a country already teetering on the brink of starvation over the edge.
The United Nations and global aid groups have cautioned that a protracted fight could force a shutdown of Hodeida's port at a time when a halt in aid risks tipping millions into starvation.
Almost 4,500 households have been displaced in Hodeida province so far this month, the United Nations said Sunday.
The Red Sea port is vital to Yemen as 80 percent of essential imports enter the country through Hodeida and many fear that any damage to the port would put millions of Yemenis at risk.
The Houthis' official news agency SABA quoted Mohammed al-Sharif, deputy head of civil aviation, as saying that images circulated online about the airport were taken in 2016 and that a fence shown as the airport fence was in fact situated on a piece of land belonging to a lawmaker.
"The armed forces which are supported by the Arab coalition have freed the al-Hodeida International Airport from the Houthi militias and the engineering teams have started to clean the airport and its surroundings from mines and bombs", the military said on its official Twitter account.
Yemeni pro-government forces gather at the south of Hodeida airport, in Yemen.
The fighting is already nearing densely populated residential areas, the Norwegian Refugee Council warned, and aid distributions have been suspended in the west of the city.
The Huthi rebels drove Yemen's government out of Sanaa in 2014, pushing President Abedrabbo Mansour Hadi into exile and sparking an intervention by Saudi Arabia and its allies the following year.
The Saudi-led coalition earlier this year imposed a near-total blockade on Hodeida port, alleging it served as a major conduit for arms smuggling to the rebels by Riyadh's regional arch rival Iran.
Mr. Dawad also said the southern gate of Hodeida city was captured by pro-coalition forces.
The United Nations says the Saudi-led assault on the Yemeni port city of Hudaydah has left almost 5000 families displaced since the beginning of this month.
Griffiths, whose talks in Sanaa have been largely kept under wraps, is believed to be pressing the Huthis to cede control of the Red Sea port to a UN-supervised committee that would allow deliveries of commercial goods and aid to continue to flow. The Houthis deny being pawns of Iran and say they are fighting to evict foreign occupiers from Yemen.
The Houthis, mountain fighters who seized Sanaa in 2014, gained valuable experience in a series of guerrilla wars with Yemen's national army and a brief border war with Saudi Arabia. Around two-thirds of the country's population of 27 million relies on aid and 8.4 million are already at risk of starving.