Saudi Arabia, Egypt, Bahrain and the United Arab Emirates sent a 13-point list of demands in return for an end to a three-week-old diplomatic and trade boycott of Doha. But Qatar rejected the demands as unrealistic.
US Secretary of State Rex Tillerson on Sunday called on Qatar and a Saudi-led group of four Arab states to sit down together in order to try to reach a resolution.
"A productive next step would be for each of the countries to sit together and continue this conversation," Tillerson said in a statement.
"We believe our allies and partners are stronger when they are working together towards one goal which we all agree is stopping terrorism and countering extremism."
Saudi Arabia, Egypt, Bahrain and the United Arab Emirates boycotted Qatar on June 5, accusing it of supporting terrorism and extremism.
The four states sent a 13-point list of demands on Thursday in return for an end to a nearly three-week-old diplomatic and trade "blockade" of Doha.
The demands included closing Al Jazeera television, curbing relations with Iran, shutting the Turkish base and paying reparations.
But Qatar on Saturday rejected the demands as unrealistic, calling the blockade "illegal."
Qatar insists that the moves against it have more to do with long-standing differences than with the fight against extremism.
"It is about limiting Qatar's sovereignty and outsourcing our foreign policy," said Sheikh Saif bin Ahmed Al-Thani, a government spokesman.
Turkish President Recep Tayyip Erdogan on Sunday dubbed the ultimatum "against international law."
Kuwait is helping mediate the dispute.
"Lowering of rhetoric"
Tillerson attempted to soothe fraying tempers in a statement Sunday, following days of telephone diplomacy with Riyadh and Doha.
"While some of the elements will be very difficult for Qatar to meet, there are significant areas which provide a basis for ongoing dialogue leading to resolution," Tillerson said.
"We believe our allies and partners are stronger when they are working together towards one goal which we all agree is stopping terrorism and countering extremism," he said.
"A lowering of rhetoric would also help ease the tension."
The diplomatic tiff, which some observers believe President Donald Trump might have encouraged through his full-throated support for Saudi Arabia during a recent visit, could threaten the future of a huge US air base in Qatar.