The dispute began in June when the four countries cut ties with Qatar, alleging the world's top liquefied-natural-gas producer uses its wealth to fund what it calls extremist groups.
The Gulf states were also unhappy with Qatar's close ties to Iran.
Qatar's foreign minister has gone to Kuwait to personally deliver a handwritten formal response from Qatar's emir to Kuwait's.
Kuwait has emerged as a mediator in the crisis that erupted last month when Saudi Arabia, the United Arab Emirates, Egypt and Bahrain cut diplomatic ties with Qatar.
At the same time, they closed their airspace and borders to the Gulf state.
They accuse Qatar of destabilising the region by supporting what they call "extremism" and "terrorism," a claim Qatar denies.
The four Arab nations also want the Qatari-funded Al Jazeera television network shut down, accusing it of giving the Muslim Brotherhood and similar groups a platform for their ideas.
And they demand Qatar hand over all individuals wanted by the four countries for terrorism, provide details about people it has funded overseas and pay an unspecified compensation.
The nations' leaders have warned of further sanctions if their demands are not met.
But Qatari defence minister Khalid bin Mohammad Al Attiyah says his country is ready to defend itself in any way necessary.
"We are ready. We stand ready to defend our country. I hope that we don't come to a stage where a military intervention is made, but we always stand ready. We are here to defend our country, and we live every day to defend our country, and we will do that."
Saudi Arabia intends to meet with the other three countries in Cairo to discuss their next moves.
But Saudi foreign minister Adel Al-Jubeir remains steadfast about the demands on Qatar.
"The Qataris have made some progress but certainly not sufficient progress to be satisfactory. There have been, also, other developments that occurred between 2014 and today, in terms of Qatar's involvement in the affairs of the neighbouring countries, in terms of its harbouring of designated financiers of terror who operate openly in Qatar, in terms of incitement and hate speech that is projected on Al Jazeera and other networks. And so the decision was made that we have to make it very clear that this is no longer acceptable."
In a phone call to both sides, United States president Donald Trump has expressed his concern about the ongoing dispute.
He says stopping terrorist financing and regional unity are both important in dealing with extremism.
Meanwhile, Al Jazeera English has used its top anchors to issue its own video message.
"To those who demand that Al Jazeera be shut down and that people's rights be suppressed, we, too, have demands. We demand journalists be able to do their jobs free from intimidation and threats. We demand journalists aren't treated as criminals. We demand ... we demand ... we demand press freedom."
Qatar has previously said the four countries' demands, including closing Al Jazeera and also ejecting Turkish troops based in Qatar, appear designed to be rejected.
The countries have not detailed any penalties to be imposed if Qatar does refuse.
But United Arab Emirates diplomats have suggested either suspending Qatar from the Gulf Cooperation Council or placing sanctions on countries continuing to trade with Qatar.
The Western-backed, six-member organisation was established in 1981 following Iran's revolution and the outbreak of the Iran-Iraq war.
It includes Saudi Arabia, the United Arab Emirates, Qatar, Kuwait, Bahrain and Omar.
