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.
Afghanistan’s 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 the holy fasting month of Ramadan.
The Taliban claimed responsibility for the attack in a statement from the terrorist group's main spokesman, Zabihullah Mujahid, on Twitter.
The attack underlined how serious the security threat facing Afghanistan remains since former Taliban leader Mullah Akhtar Mansour was killed in a US drone strike last month and was replaced by Mullah Haibatullah Akhundzada.
The blast follows a deadly suicide attack on a bus carrying justice ministry staff near Kabul last month and a separate attack on a court in the central city of Ghazni on June 1.
The Taliban claimed both those attacks in revenge for the execution of six Taliban prisoners.