Gulshan Police Sub-Inspector Saiful Islam told bdnews24.com that the accident on the Gulshan-8 road took place around 9.30 am on Monday.
The injured bicycle driver was admitted to the general hospital of Kurmitola. The police, however, could not immediately announce his name.
The police driver, who was injured by a sergeant's engine, was called Pizush. The local population got complaints because he drove a motorcycle across the road.
When asked about this, Deputy Inspector Saiful said: "Whoever comes in the opposite direction is being investigated."