A small British military reconnaissance team is in Sudan to assess evacuation options, BBC News understands.
The government is looking at “every single possible option” to get UK citizens out, minister Andrew Mitchell told MPs on Tuesday.
Hundreds have died after clashes between rival military factions broke out on 15 April, primarily in Khartoum.
On Sunday, the UK airlifted diplomats and their families out of Sudan in a special military operation.
But the government is facing growing criticism over its lack of help for the British nationals still stranded there.
Up to 4,000 UK citizens could be there and 2,000 of them have already requested help, but any evacuation carries “grave” risks, the government has warned.
Some UK nationals have said they felt abandoned by the government.
Amar Osman, a British citizen from Dunfermline in Fife, told the BBC he feared his family would die in Sudan unless they could get themselves out after becoming trapped north of the capital.
“It’s getting worse by the minute, so we’re thinking of evacuating by road to Egypt,” said Mr Osman, who was visiting relatives when fighting began.
“I’m doing it all by myself. I’m getting the money together, I’m getting all my family together. There’s six of us.”
The BBC understands a small military team has landed in Port Sudan, more than 500 miles from the capital, to asses potential options for the evacuation of UK citizens still stranded in Sudan.
No decisions have been made about extracting citizens, but defence sources say that work is under way to provide the prime minister with options.
It’s understood that two Royal Navy ships are already in the region – the frigate HMS Lancaster which was already at sea, and supply ship RFA Cardigan Bay which is in Bahrain where it has been undergoing maintenance.
Mr Mitchell said anyone trapped in Sudan should stay indoors where possible but can “exercise their own judgement about whether to relocate”, adding that they “do so at their own risk”.
He told the Commons that Khartoum’s main airport, where there has been fighting in recent days, was “out of action”. Energy and communications were disrupted, food and water were becoming “increasingly scarce”, and internet and telephone networks were becoming difficult to access, he said.
He said British nationals were being sent “at least daily updates” from the UK government, amid criticism from some who feel they have been abandoned in Sudan.
Responding to questions from MPs, Mr Mitchell confirmed that neither the UK’s ambassador to Sudan or the deputy head of mission were in the country when the conflict began.
A team of 200 officials is working around the clock in the Foreign Office to provide consular assistance to those who need it, he added.
Downing Street has confirmed that the UK is working with EU countries and the US on “shared challenges” in Sudan, with several evacuations led by other countries already under way.
An EU diplomatic source told the BBC that more than 1,100 EU citizens have now been evacuated, out of around 1,700 thought to be in the country.
British doctor Iman Abu Gargar told the BBC she was able to leave with the French evacuation because Irish passport holders, including her son, were able to join.
Speaking from Djibouti, which lies to the east of Sudan, she claimed to have seen hundreds of soldiers from other European countries but felt left behind by the UK.
Dr Gargar, who was forced to leave her father behind, said: “There were only difficult decisions to make. I hope no-one has to make the decisions I had to make.”
Foreign Secretary James Cleverly earlier warned that help for UK nationals remained “limited” in the absence of a ceasefire.
Some MPs have put pressure on the government to speed up efforts, including Alicia Kearns, the Tory chairwoman of the Commons Foreign Affairs Committee, who told the Commons that “time is running out”.
Labour’s shadow minister for Africa Lyn Brown said what people trapped in Sudan need to hear “is a clear plan on how the government will support those still in danger and how they will communicate with them and when”.
She added: “Naturally, questions will be asked about whether the government has learned the lessons of the chaotic Afghanistan withdrawal.”
A successful operation to rescue diplomats and their families was carried out over the weekend, after gun battles broke out around the embassy in Khartoum.
The BBC understands that UK special forces troops landed in Khartoum on Saturday alongside the US evacuation team.
Military vehicles were used to rescue embassy staff and transport them to an airport outside the capital, before they were flown to Cyprus.
Around 1,200 personnel from the British army, Royal Navy and Royal Air Force were involved in the rescue, and a C-130 Hercules and Airbus A400M transport aircraft were used.
Defence minister James Heappey said the embassy rescue mission “went without a hitch” despite its complexity but that the “job isn’t done”.
The Ministry of Defence is working on options to support British nationals in Sudan which will be presented to the prime minister, he added.
The situation on the ground is at times “extremely dangerous”, he said, and the “window in which the environment is permissive is rarely long enough in which to do the military options”.
Mr Heappey admitted the UK had been caught out by the rapid deterioration in Sudan, adding: “It is fair to say that nobody in the UK government nor really in the wider international community saw fighting of this ferocity breaking out in the way that it did.”
Around 400 UK nationals in Sudan hold only a British passport, while about 4,000 more are dual citizens, Mr Mitchell told the Commons, adding people would “be treated in the same way” irrespective of their status.
Another Cobra meeting – an emergency response committee made up of ministers, civil servants and others – was held on Monday to discuss the situation.
UN secretary-general Antonio Guterres told a meeting of the UN Security Council that the situation in Sudan was worsening and the country was on “the edge of the abyss”.
“The violence must stop. It risks a catastrophic conflagration within Sudan that could engulf the whole region and beyond,” he said.
Communications in Sudan have been limited due to internet blackouts. The internet monitoring group Netblocks said on Sunday that connectivity was at 2% of normal levels.
The internet in Khartoum has been down since Sunday night, according to a BBC reporter in the country earlier on Monday, amid reports that one remaining provider had been taken down by one of the groups involved in the fighting in order to stop its rival from streaming programmes onto national TV.
State TV, which has broadcast material supportive of the ruling army junta, was largely off air on Monday.