The first blast happened around 7:50a.m. on Silo road in Police District 5 of the city, said Nasrat Rahimi, a spokesman for Interior Ministry.
The incident was triggered by a magnetic bomb attached to a bus that belonged to Rabbani Education University, a public university located in PD 5, he said.
Ten people who were travelling on the bus were slightly injured as a result of the explosion, Rahimi said. One was killed, according to Wahidullah Mayar, a spokesman for Public Health Ministry.
Later, two consecutive blasts went off in the same area, injuring at least seven people, Mayar said.
Daesh claimed responsibility for the bombings.
They come two days after a car bomb attack on a US convoy in Kabul killed at least four Afghan civilians and wounded four US service members.
Nearly a week ago, at least 10 people were injured when a bus carrying employees of ministry of haj and religious affairs was bombed in the city.