A Kuwaiti news agency has reported details of Qatar's response to the demands made by the "embargo countries".
The minister added that Qatar's response to the ultimatum was conveyed to Kuwait, which acts as a mediator in the crisis.
In June, Saudi Arabia, the United Arab Emirates, Bahrain and Egypt severed diplomatic ties with Qatar, accusing it of sponsoring terrorism. The four states stated that if Qatar does not comply with the demands, the crisis with Doha will continue and they will impose additional sanctions.
The deadline to comply with the demands ended on Sunday, but the Gulf States and Egypt granted Qatar another 48 hours before the group would consider imposing further sanctions.
He also revealed that Qatar's neighbors are plotting to topple its emir and warned the Saudi-led Arab states that, historically, Qatar had proven it was not an easy country to be "swallowed up".
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, like the countries lined up against it, is a USA ally.
The minister was carrying a handwritten letter from Qatar's Emir Sheikh Tamim bin Hamad Al Thani to Kuwait's Emir Sheikh Sabah Al-Ahmad Al-Jaber Al-Sabah, Kuwait's official KUNA news agency said. The list of demands to overcome the crisis comprised 13 points, including the end of diplomatic relations with Iran and the closure of a Turkish military base in the emirate.
Al Thani was set to immediately fly back to Qatar after the meeting. He accused Qatar of supporting extremist organizations.
In a statement, the lawyers also called on the worldwide community to condemn the actions of the group of countries which they said were illegal.
Late on Sunday, before the postponment, US President Donald Trump spoke to the leaders of Saudi Arabia, Qatar and the Crown Prince of Abu Dhabi in the UAE separately to voice his "concerns about the ongoing dispute".