The explosion happened in the heavily secured downtown area near Kabul University at a time of increasing attacks by the Taliban and other insurgents around Afghanistan, where the outcome of a presidential election is still not known.
At least five of the dead were air force personnel, said Hashmat Stanekzai, a spokesman for Kabul's police chief. Stanekzai said the bus had been the bomber's target.
Heavy fighting is still underway in the strategically important Sangin district of southern Helmand province. Many Afghans fear overall security is deteriorating.
The bulk of NATO troops are due to withdraw this year and Afghan security forces have taken primary responsibility for securing the country.
The violence comes amid heightened political tensions over the disputed second round of the presidential election, Reuters reported.
One of the two candidates, former foreign minister Abdullah Abdullah, has complained that extensive vote rigging had taken place to the advantage of his rival, former World Bank official Ashraf Ghani.
Results of the second round were to have been released on Wednesday.
With President Hamid Karzai's second term coming to an end, the election is meant to represent the first democratic transfer of power in Afghanistan's turbulent history.