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.
Sayed Gul Agha Hashimi, chief of Kabul's Criminal Investigation Department, confirmed the death toll and estimated number of injured.
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.
"As a result of this attack, a number of policemen and officers were killed and wounded."
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.
A blast in central Kabul housing several embassies and NATO's military headquarters a week ago was blamed on a grenade that exploded during a routine vehicle search.
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.