The attacker targeted the bus as it left a police training centre on the Jalalabad road, a main route out of the city and a regular scene of militant strikes in recent years.
"We have two people killed -- a policeman and a civilian," Hashmat Stanikzai, Kabul city police spokesman, told AFP.
He said the suicide bomber riding a bicycle targeted the police bus and that 20 police and passers-by had also been wounded, some of them seriously.
The bus was left at the side of the road with its chassis badly damaged and most of its windows broken.
Eye-witness Abdul Majeed, a stallkeeper who was at the scene, told AFP: "I saw a bicycle rider hit the bus and then heard an explosion."
The Taliban, who have been fighting the US-backed government since 2001, claimed responsibility for the attack.
"One of our mujahideen fighters targeted a police bus belonged to the police training centre," Zabihullah Mujahid, a Taliban spokesman, said.
Kabul has seen a drop in attacks after a series of high-profile strikes in the first half of last year, with the intelligence agency claiming to have foiled several plots involving truck bombs and suicide gunmen.
A series of attacks in 2013 targeted foreign compounds, the Supreme Court, the airport and the presidential palace in the city.
NATO forces are withdrawing from Afghanistan after more than a decade of fighting the Taliban, but negotiations have stalled on a security accord that would allow some US and NATO troops to stay after 2014.
In the last major blast in Kabul, a Taliban suicide attacker detonated an explosives-packed car next to a NATO military convoy on December 27, killing three NATO personnel and injuring six civilian passers-by.
That attack was also on the Jalalabad road, which passes a series of government compounds and military facilities.
Afghanistan's fledgling security forces face a difficult year in 2014 as insurgents attempt to disrupt elections on April 5 that will choose a successor to President Hamid Karzai and as NATO's combat mission winds down by December.
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