Reuters was citing an official from one of the four countries.
In recent years especially, the Arabic-language channel has become closely associated with Qatar's promotion of the Muslim Brotherhood and other Sunni Islamists around the region, including those who have advanced extremist or sectarian views on the channel.
The four countries have not made their demands public yet.
According to the list, Qatar must refuse to naturalise citizens from the four countries and "revoke Qatari citizenship for existing nationals where such citizenship violates those countries' laws".
"These demands must be agreed within 10 days, otherwise they would be invalidated", according to the document.
The Saudi alliance severed diplomatic and transport links with Qatar on June 5, accusing their fellow Gulf Cooperation Council member of supporting terrorism.
Four of the Arab states involved in the boycott of Qatar have sent a list of demands to the country.
Qatari officials have not yet responded to CNN's requests for comment but said they would issue a statement later Friday.
Qatar's foreign minister on June 19 said his country wouldn't bargain away what it sees as its sovereign rights and called on the Saudi alliance to conduct negotiations in a "civilized way", after first lifting the blockade.
Qatar must also announce it is severing ties with terrorist, ideological and sectarian organisations including the Muslim Brotherhood, Islamic State, al-Qaeda, Hezbollah, and Jabhat Fateh al Sham, formerly al Qaeda's branch in Syria, he said, and surrender all designated terrorists on its territory.
Meanwhile, there are also five armored vehicles in the Turkish military base in Qatar.
"The brother (Qatar) must realise that the solution for its crisis lies not in Tehran or Beirut or Ankara or Western capitals or in media outlets, but in regaining the trust of its neighbours", Gargash said.
Saudi Arabia and its neighbouring Gulf States have given a list of demands to Qatar, saying they will lift the trade and diplomatic embargo if Qatar comply.
Reports have suggested countries behind the economic sanctions on Qatar - Saudi Arabia, UAE, Bahrain, Egypt, and others - are demanding the closure of Al Jazeera, a media group that has been targeted in the Middle East because of its critical reporting. For the following 10 years, Qatar would be monitored annually for compliance.
Doha has not yet issued a formal response to the demands, but government representatives have previously stated negotiations can not begin until the Arab coalition lifts its sanction regime.
Since the onset of Qatar's isolation, Turkey has fast-tracked a decision to approve the deployment of troops to Qatar - part of an existing bilateral agreement but widely interpreted as a show of support for the increasingly isolated country.