The minister added that Qatar's response to the ultimatum was conveyed to Kuwait, which acts as a mediator in the crisis.
The demands included that Qatar shut down the Al Jazeera Media Network, close a Turkish military base and scale down ties with Iran.
Qatar's foreign minister described Saudi Arabia and its allies latest set of demands against the energy-rich Kingdom as impossible to meet on Tuesday.
Envoys are working intensely to tackle the crisis that began on June 5, as the countries cut off diplomatic ties with Qatar over their allegations that Doha is funding extremist groups and destabilizing the Gulf region.
Saudi Arabia and three other Arab states which spearheaded a diplomatic campaign against Qatar have extended the deadline for Doha to accept their demands.
"There is no fear".
Gabriel said he had seen chances of that during meetings on Monday in Saudi Arabia and also in Qatar and that "international bodies" could be involved in the discussions.
State Department spokesman Heather Nauert said that the talks will last for a week but he said that Saudi Arabia's demands are still challenging for Qatar.
Qatar denies the charges that it was supporting "terrorism and extremism".
Speaking to reporters on Saturday, he said: "We believe that the world is governed by global laws that don't allow big countries to bully small countries".
Egypt, Saudi Arabia, Bahrain, and UAE chose to extend the deadline two more days for Qatar to comply with their demands.
The official KUNA news agency reported that Sheikh Mohammed bin Abdulrahman Al Thani arrived in Kuwait earlier today to deliver a message from Qatari Emir Sheikh Tamim bin Hamad Al Thani. Trump also underscored that unity in the region is critical to accomplishing the Riyadh Summit's goals of defeating "terrorism" and promoting regional stability, the White House said.
Cairo also will play host to a meeting of a United Nations agency monitoring worldwide air travel over a complaint by Qatar about its neighbors cutting off its air routes over the dispute.