WASHINGTON (CNN) -- Iranian influence on Iraq's ruling parties is a "stark reality," the commander of U.S. troops in Iraq said Thursday, but he said Iranian support for Iraqi Shiite Muslim militias has raised "genuine concern" among leaders in Baghdad.
Gen. David Petraeus, the commander of U.S. troops in Iraq, briefed Congress Tuesday.
In a CNN interview after two days before Congress, Gen. David Petraeus said some Iranian influence in Iraq was "understandable" given the ruling Shiite coalition's longstanding ties to Iran.
"But there's another reality, and that is a keen Iraqi desire for sovereignty," Petraeus said. "Iraq is a big country. It is potentially a very wealthy country. It has no desire to be the 51st state of Iran, if you will."
In two days of congressional hearings, Petraeus painted Iran's support for Shiite extremists as the biggest long-term risk to Iraq's stability. But he said the recent Iraqi government attempt to crack down on Shiite militias in Basra and Baghdad has shown that Iranian support for those groups "is of genuine concern by really all Iraqi leaders."
"You see leaders of parties that again have benefited financially, physically -- all kinds of different ways -- from their relationships with Iran now being gravely concerned about what the special groups, and to a degree, the militias are doing in Iraq," he said. Watch Patraeus discuss congressional hearings »
Iraq's Shiite-led ruling coalition is dominated by the Islamic Supreme Council of Iraq. Formerly known as the Supreme Council for the Islamic Revolution in Iraq, its leaders were sponsored by Iran during former dictator Saddam Hussein's rule -- and its Badr Corps militia fought alongside Iranian troops during the Iran-Iraq war of the 1980s.
In addition, much of Prime Minister Nuri al-Maliki's Dawa Party leadership took refuge in Iran before Hussein's overthrow by a U.S.-led army in 2003, though al-Maliki himself was in exile in Syria.
The warm welcome Iranian President Mahmoud Ahmadinejad received during his March visit to Baghdad rankled several American lawmakers during Petraeus' testimony this week.
But Petraeus said Iraqi leaders have sent "very direct" messages to Tehran in recent weeks, warning that it risks alienating friendly Iraqis if it continues to support violence. Meanwhile, he said, al-Maliki's government was drawing support across the country's sectarian and ethnic divides by confronting what it called "outlaw" militias during the crackdown it launched in March.
"Iran will have to come to grips with its own conflictions, between wanting to wish us ill and not wanting the first Shia-led Arab country to fail," Petraeus said.
U.S. troops and warplanes supported the government's effort to reclaim control of Basra. Petraeus said the "special groups" are responsible for many of the recent U.S. combat deaths, and President Bush warned Iran Thursday to stop poking at Iraq's internal divisions.
The Supreme Council now controls the country's Interior Ministry, and much of Iraq's security forces are dominated by members of the Badr Organization, as its onetime militia is now called. Badr members fought with Iraqi troops against members of the Mehdi Army, the militia led by rival Shiite leader Muqtada al-Sadr, during the Basra crackdown.
Al-Sadr's followers say they were unfairly singled out by the Basra operation and continue to wage pitched battles against Iraqi security forces in Baghdad. They accuse the government of trying to cripple their movement ahead of provincial elections in October.
The Sadrists control about 30 seats in Iraq's parliament and were instrumental in forming the ruling coalition, but broke with al-Maliki in 2007 when he refused to endorse a timetable for the withdrawal of U.S. troops.
Al-Maliki demanded this week that al-Sadr disband the Mehdi Army or see his followers barred from public office. But Petraeus said al-Sadr's movement "has to be included" in any settlement of the five-year-old war, "and has to be again treated like the important political element that it is." E-mail to a friend
CNN's Thomas Evans contributed to this report.