Neighbours including Saudi Arabia and the United Arab Emirates (UAE) have closed their airspace to Qatari planes.
Meanwhile, US President Donald Trump has said he was told during his recent visit to Saudi Arabia that Qatar was funding "radical ideology".
"During my recent trip to the Middle East I stated that there can no longer be funding of Radical Ideology. Leaders pointed to Qatar - look!" he tweeted.
Kuwait - one of the Gulf countries not involved in the dispute - has offered to mediate talks, and Qatar said it was receptive to dialogue.
In an interview with Qatar's al-Jazeera network, the Qatari foreign minister said Kuwait's emir would travel to Saudi Arabia on Tuesday for talks.
Sheikh Mohammed Bin Abdulrahman al-Thani told Al Jazeera that Qatar was seeking "a dialogue of openness and honesty".
Six countries - Bahrain, Saudi Arabia, the UAE, Yemen, Libya's eastern-based government and the Maldives - cut diplomatic ties with Qatar on Monday.
Bahrain, Saudi Arabia and the UAE have given Qatari nationals two weeks to leave, and banned their own citizens from travelling to Qatar.
What has happened?
The states that joined Monday's move against Qatar, a tiny but gas-rich peninsula, include some of the biggest powers in the Arab world.
Saudi Arabia, Bahrain and the UAE closed all transport links by air, land and sea.
The UAE and Egypt expelled Qatari diplomats, giving them 48 hours to leave, and Saudi Arabia closed down a local office of al-Jazeera.
Disruption to airspace in the Gulf began on Tuesday morning local time. Doha, Qatar's capital, is a major hub for international flight connections.
Airlines affected by the airspace restrictions include Qatar Airways, Etihad Airways and Emirates.
When avoiding Saudi Arabia, their massive - and only - neighbour, Qatar's planes are having to take more indirect routes, leading to longer flight times.
On Tuesday, Saudi Arabia revoked Qatar Airways' licence and ordered the company's Saudi offices to shut within 48 hours.
In a country reliant on imported food, residents have started to stockpile.
"People have stormed into the supermarket hoarding food, especially imported ones," one Doha resident, Eva Tobaji, told Reuters. "It's chaos - I've never seen anything like this before."
A substantial amount of Qatar's food is transported across the border from Saudi Arabia, which is being closed. That is also an important route for construction materials - needed for the energy industry and for the preparations for the 2022 football world cup.
Qatar's exports are dominated by oil and gas. They are mostly seaborne, so should not be immediately hit, but the general economic disruption could have an impact if the dispute drags on.
That possibility pushed the price of crude oil higher, but only briefly. Qatar is a member of the exporters' group Opec and the dispute could yet undermine the organisation's efforts to raise prices by restricting production.
Why has this happened?
While the severing of ties was sudden, it has not come out of the blue, as tensions have been building for years, and particularly in recent weeks.
Broadly, two key factors drove Monday's decision: Qatar's ties to Islamist groups, and to Iran, Saudi Arabia's regional rival.
Wealthy individuals in Qatar are believed to have made donations and the government has given money and weapons to hardline Islamist groups in Syria - Qatar says this is not the case.
The Financial Times also reports that Gulf allies were angry that Qatar paid a $1bn (£773m) ransom to jihadists and Iranian security officials after Qatari nationals were kidnapped in Iraq and Syria.
Saudi Arabia, too, has been accused of funding IS, either directly or by failing to prevent private donors from sending money to the group - allegations it denies.
Is Saudi to blame for IS?
Analysts also say the timing of the diplomatic withdrawal, two weeks after a visit to Riyadh by President Trump, is crucial.
Mr Trump's speech in Saudi Arabia, in which he blamed Iran for instability in the Middle East and urged Muslim countries to take the lead in combating radicalisation, is likely to have emboldened Gulf allies to act against Qatar.
In the same week as Mr Trump's speech, Egypt, Saudi Arabia, Bahrain and the UAE blocked Qatari news sites, including al-Jazeera. Comments purportedly by Qatar's Emir, Sheikh Tamim bin Hamad al-Thani, criticising Saudi Arabia had appeared on Qatari state media.
The government in Doha dismissed the comments as fake, attributing the report to a "shameful cybercrime".
The latest reaction
- In the interview with al-Jazeera, Foreign Minister Sheikh Mohammed Bin Abdulrahman al-Thani said Qatar would not retaliate but was unhappy with regional rivals "trying to impose their will on Qatar or intervene in its internal affairs"
- He said Qatar's emir, at the request of his Kuwaiti counterpart, had cancelled a speech scheduled later for Tuesday, to allow negotiations to proceed calmly
- The Philippines, which has an estimated 200,000 residents in Qatar, said it had stopped the deployment of workers to the Gulf state
- Philippines labour official Silvestre Bello told a new conference he feared Filipinos in Qatar would "be the first victims" in the event of riots over food shortages.
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