Kuwait's Emir, Sheikh Sabah Al-Ahmad Al-Jaber Al-Sabah, reportedly made the request for an extension to the 3 July deadline after receiving assurances from the Qataris that they would still deliver their response to him by Monday.
Gabriel's trip comes as Saudi Arabia, along with the United Arab Emirates, Bahrain and Egypt, have boycotted Qatar over what they say is its support for Islamist extremism and close ties to Iran, seen by Gulf states as a risky regional rival.
Qatar was given 10 days to comply with the demands but as Sunday's deadline passed, the Gulf neighbors have given the country until Tuesday to respond.
Saudi Arabia's King Salman bin Abdulaziz Al Saud will not to attend the G20 summit due to the Gulf crisis, Saudi diplomatic sources said Monday.
The Four countries, on 23 June, filed a list of 13 demands including closing Al Jazeera television, reducing ties to Iran, closing of a Turkish military base in Qatar and the handing over of all designated terrorists on its territory.
Qatar's foreign minister, carrying a handwritten letter from the country's ruling emir, has arrived in Kuwait amid a diplomatic crisis engulfing his nation.
While Trump has seemingly sided with the Saudi-led bloc, his secretary of state, Rex Tillerson, has called for an end to the diplomatic and economic blockade, which he said has hurt U.S. interests in the region.
"These tactics are created to aggressively force Qatar into an impossible position, which must be condemned by the global community".
"The meeting will discuss the future steps that will be undertaken by the four Arab countries towards Qatar", the statement read.
Egypt said it would host the foreign ministers of Saudi Arabia, the United Arab Emirates and Bahrain on Wednesday to discuss the crisis.
On the calls, Trump addressed his concerns about the ongoing dispute that has seen Saudi Arabia and other Sunni Arab states, including Egypt, United Arab Emirates and Bahrain, cut ties with Qatar since the beginning of June over its alleged support for extremist groups.
Speaking in Washington last week, the Qatari foreign minister said the GCC was set up to guard against external threats.
Qatari academician and writer Dr Abdulhamid al-Ansari said the online campaigns launched against the country indicated the political failure of the siege countries and their inability to present to the world any valid justification to impose the blockade against Qatar.