The two victims were in Gould Hall, where authorities responded at about 9 a.m. to reports of shots being fired, according to Seattle Fire Department spokeswoman Helen Fitzpatrick.
KOMO-TV reported that a woman and a man were shot to death.
Gould Hall, built in 1972, is the university's architecture building. It houses three departmental offices, a dean's office, a library, shop, lab, computer facilities and classrooms, according to the university's Web site.
Russia had already conducted its investigation into the use of chemical weapons in Douma and did not find either a single sign of the chemical attack or witnesses