The bomb apparently exploded around 0445 GMT when the vehicle was passing through the Halkali district, a suburb on the European side of the city, according to TV channels NTV and CNN-Turk, which broadcast images of the heavily damaged coach.
There were also family members of the soldiers on board the coach, including children, the TV stations reported.
Several ambulances were rushed to the scene which has been cordoned off by police.
No one immediately claimed responsibility for the attack.
The rebel Kurdistan Workers Party (PKK) threatened at the weekend to launch attacks in all Turkish cities.
The outlawed PKK, considered a terrorist group by Turkey and much of the international community, has waged an 26-year campaign against the Turkish army for a independent homeland in the country's southeast, a majority Kurdish region. The group has also in the past been responsible for attacks in the country's main cities.