The ultimatum presented to Qatar also includes demands that the natural gas-rich state stop funding a host of other news outlets including Arabi21, Al-Araby Al-Jadeed and Middle East Eye.
The demands, the official said, was handed to Qatar by mediator Kuwait.
Yemen and Egypt had announced earlier this month that they have joined Bahrain and Saudi Arabia to withdraw their diplomatic staff from Qatar.
"The leak (of the demands by Qatar) is an attempt to abort the mediation in a childish act that we have grown accustomed to from our brother", Gargash wrote on Twitter.
The list of 13 demands circulating on social media also says Qatar must cut ties to extremists including the Muslim Brotherhood, Islamic State group, Al-Qaeda and Lebanon's Iran-backed Hezbollah movement.
Reuters confirmed Turkish reports that five armoured vehicles and 23 military personnel arrived in Doha yesterday in a deployment Turkey's armed forces said was part of a military training and cooperation deal. The United States Secretary of State Rex Tillerson, on June 21, had said that he hopes the demand laid out by the bloc would be "reasonable and actionable".
Qatar's Foreign Minister Sheikh Mohammed bin Abdulrahman Al Thani has said that Al Jazeera Media Network is an "internal affair" and there will be no discussion about the fate of the Doha-based broadcaster amid the Gulf crisis.
Egypt and the boycotting countries gave Qatar 10 days to comply.
Qatar is yet to comment on the demands, although officials have previously said that they will not negotiate until the blockade is lifted.
"The leakage will further exasperate & prolong the Qatar crisis".
According to the list, Qatar must refuse to naturalize citizens from the four countries and expel those now in Qatar.
Al Jazeera Media Network is owned and funded by the Qatari royal family.
In recent years, critics have argued that it is strongly supportive of Islamists, especially Egypt's now-banned Muslim Brotherhood.
"The base in Qatar is both a Turkish base and one that will preserve the security of Qatar and the region", said Isik.
Saudi Arabia has severed all land, sea and air links with Qatar, and Qatar Airlines has had to cancel or reroute some flights that usually use the airspace of the boycotting nations.
In addition, the country must locate and extradite all individuals now facing terrorism charges in Saudi Arabia, Egypt, and the other signatory states. "Re-evaluating the base agreement with Qatar is not on our agenda".
Saudi Arabia regularly accuses Iran, its regional rival, of interference throughout the Middle East.