In a statement issued in Washington, Tillerson said some of the demands presented by Bahrain, Egypt, Saudi Arabia and the UAE will be very hard for Qatar to meet, US Secretary of State Rex Tillerson said, calling on the sides to focus on areas which provide a basis for dialogue.
"[Iran's] help for Qatar's economy and the expansion of ties, particularly between the two countries' private sectors, can be among our common goals", Rouhani said.
Saudi Arabia, the United Arab Emirates, Egypt and Bahrain imposed a boycott on June 5 on Qatar and issued 13 demands including closing Al Jazeera television, curbing relations with Iran, shutting the Turkish base and paying reparations.
Qatar is under strict sanctions from Saudi Arabia and its allies, Egypt, the UAE and Bahrain.
Doha denies the accusations, describing the attempts to isolate it as "unjustified".
The president said the 13-point demand list by Arab states, already rejected by Qatar, contradicted worldwide law.
"A productive next step would be for each of the countries to sit together and continue this conversation", Tillerson said.
Mr Tillerson is also calling for a "lowering of rhetoric" to "help ease the tension" and said the United States is supporting Kuwait's efforts to mediate. In public, Qatari officials have dismissed the list as unrealistic and disingenuous. Qatar, who has been accused of providing financing to Al-Qaeda and other Islamist groups, has rebuffed these demands saying they are an impingement of its sovereignty and have nothing to do with fighting terrorism.
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Anwar Gargash, the UAE's foreign minister, said on Saturday that Qatar "cannot be a Trojan horse" with regard to regional security. That presumably would mean Qatar would have to close down Al-Jazeera's English-language affiliate.
The Turkish parliament approved the deployment of troops to the base in Qatar only two days after the Gulf crisis broke out earlier this month.
The United States says some demands on Qatar (KUH'-tur) by its Mideast neighbors "will be very hard to meet".
Sudanese President Omar al-Bashir reiterated his support to Kuwait in bridging the gap between Gulf States and Qatar.
"We assert our right to practice our journalism professionally without bowing to pressure from any government or authority", it said.
Al-Thani reportedly hailed Iran as an "Islamic power" and criticized US President Donald Trump's policy towards Tehran on a news website.
A handout picture provided by the office of Iranian President Hassan Rouhani shows him speaking at a conference in the capital Tehran.
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