Gen. Ayub Salangi, Kabul's police chief, said the insurgents fired rocket-propelled grenades and machine guns from their positions before two of them detonated their explosive vests and the other five were killed by return fire from Afghan army and police forces supported by Norwegian special forces, The Wall Street Journal reported.
There were no casualties among the Afghan forces or civilians during the pitched 4-hour gun battle, the newspaper said.
A coalition spokesman said the attack closed Kabul airport to incoming and outbound flights. The assault came hours after President Hamid Karzai flew from the airport to Qatar for a meeting to discuss plans for opening a Taliban negotiating office there, the Journal said.
The Taliban said they staged the attack on the military portion of the airport because U.S. forces are stationed there, the Journal said. A major NATO military base is located near the airport.
The BBC, quoting a local police official, reported the insurgents fired from a five-story building under construction. Police, led by rapid reaction force personnel, entered the building and were going from floor to floor while police helicopters hovered overhead.
The incident began just after dawn Monday, with reports of explosions and gunfire.
The Washington Post reported there was a suicide bomb explosion prior to the start of the shooting. The report said the insurgents have used similar tactics in previous such attacks.
A Western official in Kabul had told the Post more assailants were involved, with 10-12 believed to be alive as of 8 a.m. Monday.
Kabul police spokesman Hashmatullah Stanekzai said the attackers entered on the west side of the airport and began fighting with security forces.
CNN, quoting police, said the attackers stormed into the building after the explosions and began their gun battle with security forces. Sirens went off in Kabul as the clashes began.
The Journal said the northern side of the Kabul airport is where the coalition's Joint Command or the war's operational headquarters is located.
NATO's International Security Assistance Force in Afghanistan said it remained under lockdown following the attack. The ISAF said it had "no operational information" on the incident.
Embassies in Kabul's diplomatic section also had locked down, with alarms sounding from U.S. and British embassies, the BBC said.
The Taliban already had announced the start of their so-called spring offensive against foreign military forces, which are due to end their combat operations by the end of next year.