President Trump late Saturday said he would not seek a quarantine on New York, New Jersey and parts of Connecticut after raising the idea earlier in the day and said his administration would instead issue a “strong travel advisory” for the area, as the nation’s largest city quickly becomes an epicenter of the coronavirus pandemic.

Mr. Trump said he had directed the Centers for Disease Control and Prevention to issue the travel advisory and that more details would be released later in the evening. He said he had made the decision...