WASHINGTON—President Bashar al-Assad of Syria has approved the use of chlorine gas in an offensive against the country’s last major rebel stronghold, U.S. officials said, raising the prospects for another retaliatory U.S. military strike as thousands try to escape what could be a decisive battle in the seven-year-old war.

In a recent discussion about Syria, people familiar with the exchange said, President Trump threatened to conduct a massive attack against Mr. Assad if he carries out a massacre in Idlib, the northwestern...