Philadelphia’s City Council is weighing a proposed 1% tax on construction to raise millions of dollars for affordable-housing programs, marking the latest push by a U.S. city to address rising residential costs.

The council could pass the new tax this week, setting up a possible showdown with Democratic Mayor Jim Kenney, whose administration warns the levy would hurt the city’s competitiveness as it vies to land Amazon.com Inc.’s second headquarters.

The...