WASHINGTON—House Democrats rushed to complete an economic package responding to the spread of the novel coronavirus and win support from the Trump administration before a scheduled weeklong break.

The package House Democrats prepared and released late Wednesday would provide for paid sick leave for people affected by the coronavirus and expand unemployment insurance, among other measures. The legislation reflects a series of priorities House Speaker Nancy Pelosi (D., Calif.) and Senate Minority Leader Chuck Schumer (D., N.Y.)...