Diplomats from Saudi Arabia, the UAE, Bahrain and Egypt told reporters at the UN they now wanted Qatar to accept six broad principles.
These include commitments to combat terrorism and extremism and to end acts of provocation and incitement.
There was no immediate comment from Qatar, which denies aiding terrorist groups.
Qatar has refused to agree to any measures that threaten its sovereignty or violate international law, and denounced the “siege” imposed by its neighbors.
The restrictions put in place six weeks ago have forced Qatar to import food by sea and air to meet the basic needs of its population of 2.7 million.
At a briefing for a group of UN correspondents in New York on July 18, diplomats from the four Gulf countries said they wanted to resolve the crisis amicably.
Saudi permanent representative Abdullah al-Mouallimi said their foreign ministers had agreed the six principles at a meeting in Cairo on July 5 and that they “should be easy for the Qataris to accept”.
According to the New York Times, they were combating terrorism and extremism, denying financing and safe havens to terrorist groups, stopping incitement to hatred and violence, and refraining from interfering in the internal affairs of other countries.
Abdullah al-Mouallimi stressed that there would be “no compromise” on the principles, but added that both sides would be able to discuss how to implement them.
The list of 13 demands handed to Qatar on June 22 included shutting down the Al Jazeera news network, closing a Turkish military base, cutting ties with the Muslim Brotherhood and downgrading relations with Iran.
Abdullah al-Mouallimi said closing Al Jazeera might not be necessary but stopping incitement to violence and hate speech was essential.
“If the only way to achieve that is by closing down Al Jazeera, fine,” he was quoted by the AP as saying.
“If we can achieve that without closing down Al Jazeera, that’s also fine. The important thing is the objective and the principle involved.”
UAE permanent representative Lana Nusseibeh warned that if Qatar was “unwilling to accept core principles around what defines terrorism or extremism in our region, it will be very difficult” for it to remain in the Gulf Co-operation Council.
Qatar has acknowledged providing assistance to Islamist groups designated as terrorist organizations by some of its neighbors, notably the Muslim Brotherhood. However, it has denied aiding jihadist groups linked to al-Qaeda or ISIS.
UAE Minister of State for International Co-operation Reem al-Hashimi said: “At this stage, the ball is in Qatar’s court.”
She added that the US had “a very constructive and very important role to play in hopefully creating a peaceful resolution to this current crisis”.
President Donald Trump was quick to claim credit for the pressure being placed on Qatar, saying it might mark the “beginning of the end to the horror of terrorism”.
However, Secretary of State Rex Tillerson questioned the list of demands, acknowledging that some elements would “be very difficult for Qatar to meet”.
Also on July 18, NBC News cited US intelligence officials as disputing a report that alleged Qatar had paid a ransom of $1 billion to Iraqi Shia Muslim militias, Iranian security officials and Sunni Muslim jihadists in Syria as part of a deal to secure the release of royal family members kidnapped in Iraq.
The officials said Qatar had handed €300 million ($345 million) in cash to Iraq’s government but that Baghdad had confiscated the money after securing the hostages’ release.
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