He said the blockade was "impairing United States and other global business activities in the region" and that the USA backed mediation efforts being pursued by Kuwait.
Qatar's Arab neighbours have blacklisted entities such as Qatar Airways over their alleged extremist connection, with Barca shirts sporting the airline's logo said to be outlawed.
The escalating crisis in the Persian Gulf erupted this week when Saudi Arabia, the UAE, Egypt and Bahrain cut ties to Qatar, accusing the country of tolerating or even encouraging support for extremist groups, including al-Qaida's Syria branch.
Barcelona have been sponsored by Qatar Airways for the past four seasons, but they have agreed a new four-year deal with Japanese online retailer Rakuten from the start of the 2017-18 campaign.
Qatar Petroleum says it has responded to a diplomatic crisis engulfing Qatar by mobilizing to mitigate the impact any action could have on its energy supplies to customers around the world.
"Fighting terrorism and extremism is no longer a choice, rather ... a commitment requiring decisive and swift action to cut off all funding sources for terrorism regardless of its financier", the official Saudi Press Agency reported, citing a government source.
Hamas welcomed Al-Thani's statements with spokesman Fawzi Barhoum saying they "expressed the noble Arab character" and Qatar's support of "the Palestinian people and its legitimate resistance".More news: OPEC and non-OPEC nations poised to extend output cuts
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On Saturday, Saudi Arabia, Bahrain and the UAE made no mention of Mr Tillerson's call.
Several other Muslim nations also downgraded their diplomatic ties with Qatar.
USA officials insisted the two men were sending the same message with different emphases, aimed at encouraging their Arab allies to put aside grievances and focus on fighting terrorism. "Process can only work following clear indication that Qatar will stop support & finance of extremism & terrorism".
Russian Foreign Minister Sergey Lavrov met with his Qatari counterpart, Sheikh Mohammed bin Abdulrahman Al-Thani, in the Russian capital. "However, we can not be pleased with the situation, where these relations between our counterparts are aggravating".
Erdogan said he had never witnessed Doha supporting "terrorism", adding Turkey "will continue to give all kinds of support to Qatar" during the economic sanctions.
Qatar's foreign minister fired back that there was "no clarity" in such accusations, speaking in an interview with RT Arabic in Moscow after emerging from talks with Russian Foreign Minister Sergei Lavrov.