The Taliban claimed responsibility for the attack in text messages to reporters, the Los Angeles Times said.
The bomber struck as soldiers were boarding the bus in the Pul-e-Soktha area of Kabul, a statement from Kabul's police chief said.
The attack followed a shooting Sunday in Kabul, where Afghan intelligence officers killed a suspected suicide bomber attempting to detonate a car bomb near the National Directorate of Security offices, security officials said.
The incidents signaled the Taliban's intent to keep up attacks on targets in the heart of the capital, even with about 100,000 coalition troops still in Afghanistan, the newspaper said.
Notable deaths of 2014 [PHOTOS]