Riyadh demanded the closure of the pan-Arab broadcaster as well as the Brookings Doha Center and the Arab Center for Research and Policy Studies, the source said on condition of anonymity.
After the reportedly heated March 5 GCC meeting, Saudi Arabia, Bahrain and the United Arab Emirates recalled their ambassadors from fellow member Qatar, which they accuse of interfering in their internal affairs and supporting the Muslim Brotherhood.
The source said Saudi Foreign Minister Saud al-Faisal had demanded three things of Doha -- "to close the (Qatari-owned) Al-Jazeera network, which stirs sedition; close the research centres in Doha, and turn over all outlaws" on its territory.
Doha's foreign minister replied that the demand constituted "interference in Qatar's internal affairs," the same source said.
Gulf officials do not usually comment on closed-door meetings.
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Qatar is seen as a supporter of the Muslim Brotherhood and its affiliates across the region, which are banned in most Gulf states.
Saudi Arabia and other Gulf monarchies have long been hostile toward the Brotherhood, fearing that its brand of grass-roots activism and political Islam could undermine their authority.
Saudi Arabia and the other two states accused Doha of giving refuge to opposition figures and of even giving some of them citizenship.
Critics have long accused the influential pan-Arab broadcaster Al-Jazeera of biased coverage in favour of the Brotherhood, and several of its journalists are on trial in Egypt for allegedly supporting the group.
Most Gulf states hailed the Egyptian military's July overthrow of president Mohamed Morsi -- a former senior member of the Muslim Brotherhood -- and pledged billions in aid. Qatar, which had strongly supported him, has seen its influence in Cairo evaporate.
The Gulf Cooperation Council includes Saudi Arabia, Kuwait, Bahrain, the United Arab Emirates, Qatar and Oman.