Qatar reportedly has 10 days to meet the list of conditions.
As well as shutting down Al Jazeera, which Qatar has previously said it would not do, the boycotting five Arab countries also want it to cut diplomatic ties with Iran.
The demands call on Qatar to end its alleged support for organisations including Hezbollah, the Muslim Brotherhood, Hamas and Islamic State.
It comes nearly three weeks after Saudi Arabia, Bahrain, Yemen, the United Arab Emirates and Egypt imposed a tight blockade on Qatar, their neighbour in the Gulf.
They accuse the wealthy peninsula of funding extremism and fomenting instability in the region.
Qatar denies that it supports groups such as IS and al Qaeda, although it recognises that members of groups such as Hamas are present in its territory.
Saudi Arabia and other states closed land borders with Qatar on 5 June, expelling its citizens from their territory and banning its national carrier from its airspace.
The escalation was the culmination of tensions that have been simmering for decades.
Saudi Arabia has itself been accused of supporting extremist organisations, but it fiercely opposes interests like the Muslim Brotherhood and Iran, which enjoy support from Qatar.
The list of demands, which was presented on Friday by mediator Kuwait, states conditions that would align Qatar more closely with the goals of Saudi Arabia.
It also insisted that Qatar expel citizens of the four countries from its territory, hand over any individuals suspected of terrorism, and provide detailed information about opposition figures that it has funded in other states.
An unspecified sum in compensation was also requested, and as well as an agreement to shut down Turkey’s military presence in Qatar.
If Qatar agreed to the conditions its compliance would be closely monitored for 10 years.
The current crisis followed a visit to Saudi Arabia by Donald Trump. The US administration has echoed Saudi allegations while urging a swift conclusion to the crisis.
Turkey has continued to support Qatar throughout the boycott, sending troops to the country, and Iran has sent food supplies.
Qatar is yet to comment on the demands, although officials have previously said that they will not negotiate until the blockade is lifted.