Due to coalition bombing that targeted the Houthi rebels, "the airport is no longer able to receive aircraft operated by the United Nations or international humanitarian organisations", the official told AFP.
Flights into Sanaa airport have largely ceased because of a Saudi-led blockade since August 2016, but there have been exemptions for aid flights that are a lifeline for the population.
The seven-year conflict has led to what the UN calls the world's worst humanitarian crisis. More than 80 percent of the population of around 30 million require aid.
The airport official, who asked not to be identified, said the United Nations should secure a halt to the raids so that the airport could resume operations.
Farhan Haq, deputy spokesman for UN Secretary-General Antonio Guterres, called "on all parties to keep the airport open for humanitarian operations" but also, more broadly, "for the airport to be open for regular civilian and commercial flights."
On Monday evening, the coalition said it had carried out "a limited number of precision strikes on legitimate military targets in Sanaa international airport".
"The operation was mounted in response to the threat and use of airport infrastructure to carry out cross-border attacks," it said in a statement carried by the official Saudi Press Agency.
Its aircraft hit six targets in the airport, including sites used to "control attacks by drones loaded with explosives" or to "train terrorist elements" for such operations, the statement said.
'Keep airport 'out of crossfire'
The Iran-backed rebels have repeatedly launched missile and drone strikes against neighbouring Saudi Arabia, targeting the kingdom's airports and oil infrastructure.
They have intensified their strikes on the kingdom in recent months.
The coalition has insisted that its strikes were "in accordance with international humanitarian law" and should have no impact on the airport's operational capacity.
Khaled al-Shayef, Sanaa airport's director general, told AFP that the "health quarantine quarters and warehouses to store export and import goods were destroyed".
A metal hangar and cement structures near an air traffic control tower were also reduced to rubble, an AFP correspondent reported.
"A UN team is on the ground at Sanaa airport to verify the extent of any damage," a World Food Programme spokesperson told AFP.
Another aid group, the Norwegian Refugee Council, urged both sides to work with the UN to reopen the airport for humanitarian and commercial flights.
"Aid delivery to the airport is now at a standstill. We urge both the authorities in Sanaa and the Saudi-led coalition to keep Sanaa airport out of the crossfire and to ensure that it can function again for medical and commercial flights," NRC country director Erin Hutchinson said.
On Tuesday the coalition said Saudi airports were prepared to receive Yemen-bound humanitarian flights and to deliver aid through "access points" under UN supervision, according to statement carried by state-owned Al-Ekhbariya television.
It claimed the Huthis had halted UN aid flights into Sanaa airport on December 19.
Saudi Arabia and its allies intervened in the Yemeni civil war in 2015, after the rebels seized Sanaa in late 2014.
Tens of thousands of people -- most of them civilians -- have been killed in the war, whose impact has also been felt in Saudi Arabia through rebel strikes.
In August, a drone hit the kingdom's Abha international airport, wounding eight people and damaging a civilian plane.
Two months later, Saudi state media said 10 people were hurt in a drone attack on a civilian airport in the Saudi Red Sea city of Jeddah.
While the UN and Washington are pushing for an end to the war, the Huthis have demanded an the coalition air blockade of Sanaa airport end before any ceasefire or negotiations.
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