During a White House meeting on March 12, Mr. Esper and General Milley argued for a more limited response to the rocket attacks — a view that prevailed on Mr. Trump, who ordered nighttime raids on five suspected weapons depots in Iraq used by Kataib Hezbollah.

Several American officials said there was an increased urgency in planning attack options against Kataib Hezbollah as the group, perhaps along with other Shiite militias, has threatened to ramp up strikes against U.S. troops stationed on Iraqi bases after the celebrations for Nowruz, the Iranian New Year, end soon. American military intelligence agencies have detected signs that big attacks could be in the works, according to a senior U.S. military official who has been briefed on some of the contingency planning in Iraq.

Kataib Hezbollah, in a statement on Wednesday, warned its fighters to prepare for possible attacks from the United States, and threatened to retaliate against Americans and any Iraqis who help them. “We will respond with full force to all their military, security, and economic facilities,” said the statement, according to SITE, a private company that monitors jihadists’ websites and postings.

The immediate targets of a Pentagon campaign against Kataib Hezbollah most likely would be the group’s leadership, bases and weapons depots, Mr. Knights said. In addition to a vast array of rockets, the group is believed to have access to a hidden arsenal of short-range ballistic missiles funneled into Iraq by Iran over the past several months, according to American intelligence and military officials.

An extended campaign could hit militia targets across a wide swath of Iraq and Syria, and possibly other Shiite militias in Iraq that are loosely aligned with Kataib Hezbollah. “You can’t just hit rank-and-file fighters, you’d have to hit leadership, most of whom have probably dispersed,” Mr. Knights said.