New York City’s emergency management office plans to build a 350-bed facility at the Billie Jean King National Tennis Center in Flushing Meadows-Corona Park as efforts intensify to supplement hospital space as the U.S. battles the coronavirus pandemic.

Construction could begin as early as Tuesday at an indoor training center at the facility, which has multiple courts and wide spaces, according to a spokesman for the U.S. Tennis Association. The beds will likely be for patients who don’t have Covid-19, the illness caused by...