The Taliban said in a telephone call to AFP that they had launched an assault, including a suicide bombing on the airport complex, which contains a major NATO base. Militants had entered the airport, they said.
An Afghan government official confirmed that clashes had taken place within the airport complex.
A guard said that after the initial huge explosion the airport had come under fire from rocket-propelled grenades, mortars and small arms. There was no immediate word on casualties.
The hardline Taliban Islamists have waged an 11-year insurgency against the Afghan government, which is backed by 100,000 NATO troops, since being overthrown in a US-led invasion for harbouring al-Qaeda leader Osama bin Laden.
A spokesman for NATO's International Security Assistance Force confirmed that there had been "multiple explosions in the vicinity of Jalalabad airport", but had no further details.