WASHINGTON—The House passed a roughly $8.3 billion emergency spending package for combating the coronavirus outbreak, sending the legislation to the Senate as lawmakers raced to respond to the quickly spreading outbreak.

The bill provides more than $3 billion for developing treatments for the virus and allocates $2.2 billion for the Centers for Disease Control and Prevention to contain the outbreak, among other measures. Under the legislation, which the Senate will also likely pass this week, more than $1 billion will go...