According to the Houthi-run Al-Masirah television channel, the Sammad-3 drone launched three attacks on the airport on Thursday.
It was not immediately clear if there was any damage or casualties.
Abu Dhabi airport tweeted earlier in the day there had been an incident involving a supply vehicle that had not affected operations. It was unclear if it was related to the reported drone attack.
An identified UAE official told Reuters news agency the attack did not occur.
A Houthi military source said the armed drone flew 1,500km before it reached Abu Dhabi’s airport.
General Abdullah al-Jafri, a spokesman for the Houthis, said the drone attack showed the movement was capable of launching attacks against vital civilian infrastructure of the Saudi-Emirati-led military coalition battling the rebels in Yemen.
“Our attack on Abu Dhabi airport shows our forces are no paper tiger like our enemies claim,” Jafri told Al-Masirah TV on the phone.
“They mocked us before, but let me make it clear that the next stage will be targeting the infrastructure of our enemies in Saudi and the UAE.”
The UAE official denied the airport attack.
“Operations at the airport are business as usual,” the unnamed official was quoted as saying.
Despite the denial, people on social media noted many flights at the airport had been delayed.
Houthi spokesman Mohammed Abdul-Salam told Al Jazeera Arabic by phone the UAE denial was “baseless”.
“It is a lie. They cannot deny the new reality on the ground,” said Abdul-Salam.
“We don’t understand the hype when it comes to our attacks. We are in a state of war. We are being attacked every day. Our people are being slaughtered every single day. Our cities, our airports are being targeted by the Saudi-UAE coalition. So, why are they are surprised by us attacking their positions?”
The purported drone attack came a day after the rebels targeted two Saudi oil tankers in the Red Sea, prompting Riyadh to temporarily suspend the transport of oil supplies through the Bab al-Mandeb Strait.
The Houthis control much of northern Yemen and have said Abu Dhabi, a member of the Western-backed coalition fighting against them since 2015, was a target for their missiles.
The UAE has an advanced anti-missile interception system – the Theater High Altitude Area Defense (THAAD) – which is designed to destroy short and intermediate-range missiles.
Andreas Krieg, assistant professor at the Defence Studies department, Kings College London, expressed scepticism over the supposed attack.
“This is quite surprising if not shocking news,” Krieg told Al Jazeera. “I haven’t seen any images yet, but you need to imagine that they must have flown 1,350km from Sanaa or somewhere else in Yemen all the way to Abu Dhabi.
“It would be shocking if it turned out to be true, considering that the UAE has such good defensive capabilities. I mean it’s very hard for me at this point to imagine this is actually possible.”
Krieg said some Houthi drones had been developed locally with the help of Iran or other external supporters. Iran and the Houthis have denied any such cooperation.
“The issue here is that the Houthis have made a lot of claims that they have new unmanned air vehicles in place, but we’ve never seen it and we’ve never seen it in use,” he said.
The United Nations has called the war in Yemen the world’s worst humanitarian disaster.
At least 10,000 people have been killed and thousands of others wounded in the conflict in Yemen, one of the world’s most impoverished nations.
More than 20 million Yemenis need aid, millions are on the brink of famine, and hundreds of thousands have been infected with diseases, mostly cholera.