Riyadh - US Secretary of State Rex Tillerson is to launch a fresh bid on Sunday to ease a crisis between Riyadh and Doha, both allies of Washington, but without high hopes of a breakthrough.Apart from the months-long crisis, Iran's rising influence in the Middle East is also expected to figure high on the agenda of America's top diplomat during talks in the two capitals.Saudi Arabia, Bahrain, the United Arab Emirates and Egypt cut ties with Qatar and imposed an embargo in June, accusing it of supporting terrorism and cosying up to Iran.Doha denies the charges and has rejected their terms for a settlement.READ: Trump's jibe deepens feud with TillersonTillerson made an unsuccessful attempt to resolve the dispute during a trip to the region in July.US President Donald Trump, after initially appearing to support the effort to isolate Qatar, has called for mediation and recently predicted a rapid end to the crisis.Counter-terrorismBut before he arrived at Riyadh's King Salman air base on Saturday, Tillerson indicated there had been little progress."I do not have a lot of expectations for it being resolved anytime soon," he said in an interview with financial news agency Bloomberg."There seems to be a real unwillingness on the part of some of the parties to want to engage."Aside from the Gulf dispute and Iran, the conflict in Yemen and counter-terrorism will also figure in his talks, the State Department said.While in Riyadh, Tillerson will also take part in the first meeting of a Saudi-Iraqi co-ordination council.Iraqi Prime Minister Haider al-Abadi is leading a top-level ministerial delegation at the meeting, in a sign of warming ties as Sunni-ruled Saudi Arabia seeks to counter Tehran's influence in Shi'ite-majority Iraq.On the Gulf crisis, the goal will be to try to persuade the two sides to at least open a dialogue.After holding a working dinner with his Saudi counterpart Adel al-Jubeir on Saturday night, Tillerson was scheduled to meet other Saudi leaders on Sunday before heading for Doha.Simon Henderson, a veteran of the region now at the Washington Institute of Near East Policy, said the disputing parties do not want to lose face."Tillerson will say: 'Come on kids, grow up and wind down your absurd demands. And let's work on a compromise on your basic differences,'" he said.Kuwait has tried to serve at a mediator, with US support, but the parties have yet to sit down face-to-face.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.