Sunni Iraqi leaders are now the most nervous about a possible withdrawal of American troops, considered a counterweight to Iran, AFP reported.
Tensions between Washington and Tehran have boiled over onto Iraqi soil this month, with the US killing top Iranian general Qasem Soleimani in Baghdad and Iran striking back at an Iraqi base hosting American soldiers.
Furious at the US hit, Iraq's parliament held a vote on January 5 to oust all foreign troops, including some 5,200 American soldiers deployed alongside local forces.
All Kurdish and most Sunni MPs boycotted the session, despite threats by Shiite factions that the minorities would be considered "traitors" for backing the US presence.
Before the vote, speaker of parliament Mohammed Halbusi, a Sunni from the western Anbar province, made an impassioned appeal to MPs to reconsider.
"The decision we take now, we may not be able to amend in an hour's time," said Halbusi, addressing the mainly Shiite lawmakers in attendance.
"The US doesn't concern me. Iran doesn't concern me. Nothing concerns me as much as Iraq does," he said.
Both Iran and Iraq are Shiite-majority countries.
Tehran has cultivated close ties with Baghdad's ruling elite for decades.
It now exercises particularly strong sway through Iraq's Hashed al-Shaabi, a network of armed groups absorbed into the state security forces that has significant parliamentary representation.
Iran's growing clout has come at the expense of the US, which led the invasion that toppled Saddam Hussein whose dictatorial Sunni-dominated regime oppressed Shiites.
That unleashed a deadly insurgency, with Sunni militias attacking US forces, before a broad Iraqi sectarian war erupted in 2006-2007 costing thousands of lives.