OTTAWA—Armed with a multibillion-dollar war chest, Canada is offering money to cities to ditch zoning restrictions that thwart residential construction as the country deals with an acute housing shortage.

Canada’s Liberal government is targeting municipal-government rules that, among other things, limit the number of units and stories per lot in a bid to increase density in the country’s cities. The effort highlights the urgency to ease a housing crunch that is fueling historical rent increases, and causing hardship among younger Canadians looking to own a home, according to polling.