“I think this administration wants to be more engaged in fighting terrorism,” Mr. Abadi added. “I sense a difference in terms of being head-to-head with terrorism.”
Neither American nor Iraqi officials, however, explained what economic support might be provided by the United States and the international community to help rebuild Iraqi cities that have been damaged during the conflict.
Nor did they explain what the American role might be after Iraqi forces retake Mosul, Iraq’s second largest city, and evict Islamic State militants from other towns.
There is broad recognition among Iraqi and American security experts that there will be a continued need to train Iraqi forces, and perhaps even conduct commando operations, if the Islamic State, also known as ISIS or ISIL, loses its so-called caliphate because any surviving militants are expected to maintain their yearslong drumbeat of terrorist bombings.
Mr. Trump hinted at the need for future American presence in Iraq by criticizing his predecessor, former President Barack Obama, for failing to negotiate an agreement that would have enabled American forces to stay. American ground troops withdrew from Iraq in 2011, as required under a security agreement brokered in 2008 by President George W. Bush.
“Certainly, we shouldn’t have left. We should never ever have left,” Mr. Trump said. “A vacuum was created, and we discussed what happened.”
Mr. Abadi volunteered little on the matter, which remains a delicate issue in Iraq and, especially, with its neighbor Iran. Asked if he had been briefed on the strategy the Trump administration is working on to defeat the Islamic State, the prime minister said that “I haven’t seen a full plan.”
James Jeffrey, a former American ambassador to Baghdad, said the coming destruction of the Islamic State, as a caliphate and fighting force, would make the administration confront difficult questions about how deeply to get involved in Iraq’s reconstruction and stabilization, and what additional political reforms might be needed to ensure that the country’s politics do not become a breeding ground for the rise of another militant group.
“The main reason we’re engaging with Iraq is combating ISIS, in the short run,” Mr. Jeffrey said in an interview. “But underneath that is the question, ‘How are we going to relate to Iraq?’”
That was also addressed in a letter to Mr. Trump from more than a dozen senators, including Bob Corker, Republican of Tennessee, who leads the Foreign Relations Committee.
“Iraq’s challenges will not be solved when ISIS is defeated on the battlefield,” wrote the senators, who argued that bringing stability to Iraq would require more power-sharing with the Sunnis and progress in resolving tensions with the Kurds. “If Prime Minister al-Abadi commits to lead Iraq along these lines, he should have our full support in this endeavor.”
Some senior members of Mr. Trump’s administration, including Defense Secretary Jim Mattis and H. R. McMaster, the national security adviser, served in Iraq.
They joined the White House meeting along with other ranking officials, including Vice President Mike Pence, Secretary of State Rex W. Tillerson and Jared Kushner, the president’s senior adviser.
Mr. Abadi brought his foreign, defense and oil ministers. Fuad Hussein, a senior official from the Kurdish autonomous region in northern Iraq, was also present, in a gesture of comity to the Kurds.
The session was held virtually 14 years to the day after Mr. Bush announced the start of the United States-led invasion of Iraq, which Mr. Trump initially supported but later opposed. Mr. Abadi spoke in classical Arabic at the start of his White House meeting. After reciting a section from the Quran, he stressed the desire for more cooperation with the United States.
But the Iraqi prime minister, who lived in exile in Britain during Saddam Hussein’s years in power, switched to fluent English in his appearance at the Institute of Peace, where he sought to assure his audience that Iraq would not be unduly influenced by Iran.
“Iraq is not under the influence of any other country,” he said. “We are looking after our own interests.”
An unintended moment of levity came when the Iraqi leader was asked about proposals that Nineveh Province be turned into a semiautonomous region after Mosul is retaken. The idea has been promoted by province’s former governor, but the notion of giving that degree of autonomy to a largely Sunni but ethnically diverse region has drawn opposition from Shiite-led Baghdad.
“We have to build bridges with others and work with others to be more secure,” he said. “Otherwise, what do you do? You build walls.”
The room erupted into laughter, and Mr. Abadi grinned, as well.
Iraq’s future will be on the agenda again this week when Mr. Tillerson convenes a 68-nation gathering of the coalition that is fighting the Islamic State. Mr. Abadi is staying in Washington to attend that session.
