A Reuters witness saw several apparently dead victims and at least two wounded being carried out of the yellow bus while police and emergency service vehicles surrounded the scene in the Banae district of the city.
The attack was the latest in a recent surge of violence that highlights the challenges faced by the Afghan government in Kabul and its Western backers as Washington slowly draws down its remaining troops despite a persistent insurgency.
Interior Ministry spokesman Sediq Sediqqi said on Twitter that 14 people had been killed and eight wounded. Police were working to identify the victims, he said.
The casualties appeared to include both Afghans and Nepalese security contractors, Kabul police chief Abdul Rahman Rahimi said.
He said the suicide bomber had waited near the compound housing the security contractors and struck as the vehicle moved through early morning traffic. As well as the bus passengers, several people in an adjacent market were also wounded in the attack, which occurred during Ramadan, Islam`s holy month.
The Taliban claimed responsibility for the attack in a statement from the Islamist group`s main spokesman, Zabihullah Mujahid, on Twitter.