Scores of shuttle buses take government and military personnel to work every morning in the capital, and they have often been targeted by insurgents despite efforts by security forces to provide better protection.
On Wednesday, Taliban suicide bombers killed at least seven people in twin attacks on two buses.
The militant group, which claimed responsibility for all three blasts, strongly oppose the US troop agreement that was signed on Tuesday.
"I can confirm a suicide attack on a bus carrying Afghan army staff," Dawlat Waziri, deputy spokesperson for the ministry of defence, told AFP. "We have three killed and eight wounded.
"The wounded were immediately taken to the hospital, and they are in a stable condition."
A police statement confirmed the death toll and injured.
Taliban spokesperson Zabihullah Mujahid said in a text message: "Around 7:00 am, a suicide attacker targeted a bus carrying Afghan security forces, as a result, a number of officers were killed and wounded."
Afghanistan and the United States on Tuesday signed the long-delayed bilateral security agreement (BSA) to allow about 10,000 US troops to stay in the country next year.
The signing took place on newly-inaugurated President Ashraf Ghani's first day in office and represented a major step towards mending frayed ties between Kabul and Washington.
After Wednesday's blasts, the Taliban said they were sending "a clear message to the stooge government that signed the slave pact, and we will step up our attacks after this".
The BSA was signed by Afghanistan's new "national unity government", which was formed under a US-brokered agreement to solve months of disputes over who won the fraud-tainted presidential election.
Ghani, who was declared the election winner, agreed to a power-sharing deal with his poll rival Abdullah Abdullah, who has taken up the new role of chief executive.
Hamid Karzai, who stood down as president on Monday, had refused to sign the BSA, underlining rocky Afghan-US relations over recent years.
Nato's follow-up mission, which will take over on January 1, will be made up of 9,800 US troops and about 3,000 soldiers from Germany, Italy and other member nations.
US-led Nato combat operations in Afghanistan will finish at the end of this year, and the Taliban have launched a series of recent offensives that have severely tested Afghan soldiers and police.
The new mission - named Resolute Support - will focus on supporting Afghan forces as they take on the militants, in parallel with US counter-terrorism operations.
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