Kennewick Police Chief Ken Hohenberg confirmed the death at a Friday news conference.
KNDU reports that police have identified the man as 32-year-old Aaron Thomas Wright of Kennewick.
Police say the man their officers were pursuing was seen pointing a gun at his own head several times during the Friday morning chase.
Officials say the man got out of the car with a gun at an apartment and pointed the weapon at police, who fired at him. He ran into the apartment but came out later, still armed with the gun. Police say several officers fired, hitting the man multiple times.
The chief says the man died at a hospital.
The shooting is under investigation.