Foreign Minister Sheikh Mohammed bin Abdulrahman al-Thani said Qatar had not yet been presented with a list of demands by countries that cut off diplomatic and transport ties.
In a readout of the call, the White House said that the two leaders discussed the critical goals of preventing the financing of terrorist organizations and eliminating the promotion of extremism by any nation in the region.
The announcement increases pressure on Qatar in a diplomatic and economic campaign to isolate the small Gulf Arab state, which is a critical global supplier of gas and hosts the biggest US military base in the Middle East.
It wasn't immediately clear how Trump's sharp condemnation might affect U.S. cooperation with Qatar, which hosts some 10,000 United States troops and a major USA air base that serves as a staging ground for operations in Syria, Iraq and Afghanistan.
A separate SPA report on Saturday acknowledged Tillerson's call for Qatar to curtail support for terrorism, without mentioning that he had also said the crisis was hurting ordinary Qataris, impairing business and harming the USA fight against the Islamic State militant group.
Trump, meanwhile, appeared to take credit for the decision of the countries to cut diplomatic relations with Qatar, putting his stamp of approval on the move despite Pentagon and US State Department attempts to remain neutral.
Those breaking ties with Qatar are the UAE, Saudi Arabia, Egypt, Bahrain, the Maldives, Mauritania and Libya's eastern-based government. "We are a platform for peace", he told reporters in Doha. In recent months, suspected Russian cyberactivities, including the use of fake news stories, have turned up amid elections in France, Germany and other countries.
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UAE Minister of State for Foreign Affairs Anwar Gargash also told Reuters there would be more curbs if necessary and said Qatar needed to make iron-clad commitments to change what critics say is a policy on funding militants.
The neighbours also resent Qatar's ties to the Muslim Brotherhood movement and its sponsorship of the pan-Arab satellite television network Al-Jazeera, which has given opposition figures a platform.
Qatar, a major player in the global gas industry and host to the biggest US military base in the Middle East, has crafted an independently-minded foreign policy, attracting criticism for engaging with groups from Hezbollah to Hamas.
"The Ministry said the investigation team confirmed that the piracy process had used high techniques and innovative methods by exploiting an electronic gap on the website of the Qatar News Agency", the foreign ministry said in a statement. Riyadh denied any involvement.
The present tension in the Gulf started right after Trump left the region.
The supplies would be brought in through Qatar Airways cargo flights, the report added. Trump's visit, CBS News' Margaret Brennan reported. But whatever the result, Iran should perhaps be warned that if this is what these Arab states are willing to do to their brother, how far would they go to hurt an enemy?
The United Arab Emirates praised Trump's "leadership in challenging Qatar's troubling support for extremism" in a separate statement released on Friday.
"This ruling applies to all airlines flying into the UAE", the spokesman said in the statement.
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