- Tillerson's visit to the Gulf, his third as secretary of state, also aims at persuading Qatar and a rival Saudi led-alliance to open the door to dialogue -- a goal he said had come to a deadlock Sunday.
Doha, Qatar | AFP | Top US diplomat Rex Tillerson pursued efforts to curb Tehran's influence in talks with his country's Gulf allies Sunday, demanding that Iran pare down its involvement in Iraq as the fight against the Islamic State group draws to a close.
Tillerson's visit to the Gulf, his third as secretary of state, also aims at persuading Qatar and a rival Saudi led-alliance to open the door to dialogue -- a goal he said had come to a deadlock Sunday.
But it is the question of Iranian influence in the region -- including in Iraq and Qatar -- that is at the centre of his visit to Riyadh and Doha, just weeks after US President Donald Trump refused to certify the Iran nuclear deal and declared an aggressive new strategy against Tehran.
The US secretary of state appears focused on boosting Sunni-ruled Saudi Arabia's clout in Iraq, where Shiite forces backed by Tehran are fighting in the north, as part of a wider regional battle for influence that extends from Syria to Yemen.
"Certainly Iranian militias that are in Iraq, now that the fighting against (the Islamic State group) is coming to a close, those militias need to go home," Tillerson said at a press conference in Riyadh.
"All foreign fighters need to go home."
The Popular Mobilisation Forces (PMF) -- powerful paramilitary units dominated by Iranian-trained Shiite militias -- have been part of the fight against IS and continue to battle different factions in Iraq.
But Tillerson's remarks were also aimed at Iran's elite Revolutionary Guards (IRGC) and their foreign operations wing, the Quds Force, according to a senior US official.
"The position of the Iraqi government and the position of our government is that there should be a single Iraqi security force answerable to the Iraqi state," the official said on condition of anonymity.
"What happens to the PMFs is they go home or they integrate into the Iraqi security forces."
Iran quickly hit back, saying it had played a crucial role in the fight against IS.
"If it wasn't for the sacrifices of the Islamic Republic of Iran ... Daesh (IS) would have installed its government in Damascus, Baghdad and (the Iraqi Kurdish regional capital) Arbil by now," Foreign Minister Mohammad Javad Zarif said during a visit to South Africa.
- 'Not ready to talk' -
Tillerson sat in on the first meeting of a joint Saudi-Iraqi coordination council in Riyadh Sunday, which Iraqi Prime Minister Haider al-Abadi hailed as an "important step toward enhancing relations".
Saudi Arabia cut ties with Iraq in 1990 following Saddam Hussein's invasion of Kuwait, but Riyadh and Baghdad have rekindled diplomatic and commercial relations this year.
Tillerson's visit comes nearly five months after Saudi Arabia, Bahrain, the United Arab Emirates and Egypt cut ties with Qatar and imposed an embargo on Doha, accusing it of supporting terrorism and cosying up to Iran.
Qatar denies the charges and has rejected their terms for a settlement.
Tillerson has refused to take sides in the crisis and said that talks between the feuding Gulf states remained unlikely.
"We cannot force talks among people who are not ready to talk," the US diplomat said at a press conference in the Qatari capital on Sunday night.
"There is not a strong indication that parties are ready to talk yet," he added. "We cannot and will not impose a solution on anyone."
Tillerson, a former chief executive of energy giant ExxonMobil, had also launched an unsuccessful attempt to resolve the dispute during a trip to the region in July.
"The United States maintains very strong relationships with all the countries involved in the Qatar dispute, including Qatar, and the United States intends to maintain those very strong, positive important relationships," Tillerson said Sunday in Riyadh.
"They're important from a security standpoint. They're important from an economic standpoint."
Trump had initially appeared to support Saudi Arabia in isolating Qatar but has since called for mediation, predicting a rapid end to the crisis.
The United States continues to support the efforts of Kuwait, which has tried to serve as a mediator in the worst diplomatic crisis to hit the Gulf in decades.
The State Department has said the conflict in Yemen, where Saudi Arabia and Iran back warring parties, would also figure in Tillerson's talks in the Gulf.
During his trip Tillerson is also to visit New Delhi in order to build what he said in a recent speech could be a 100-year "strategic partnership" with India.
Tillerson will stop in Islamabad to try to sooth Pakistani fears about this Indian outreach, but also pressure the government to crack down harder on Islamist militant groups.