The first arrivals boarded the vessel moored at Portland in Dorset on Monday. Pictures appeared to show two men being escorted on to the barge by staff in high-vis jackets, while a coach was also seen arriving at the port.
Speaking earlier in the day, the Home Office's Sarah Dines confirmed “all possibilities” were being studied, after reports the government was considering reviving plans to fly people who arrive by unauthorised means to a small British Overseas Territory in the South Atlantic.
Asked whether all 500 planned asylum seekers could be on board the three-storey barge by the end of the week, Ms Dines told BBC Radio 4's Today programme: “Yes, quite possibly it will be 500, we are hoping.”
Steve Smith, Chief Executive of refugee charity Care4Calais told The National none of the asylum seekers the charity is supporting boarded the barge on Monday due to successful campaigning on their behalf.
He said: “None of the asylum seekers we are supporting have gone to the Bibby Stockholm today as legal representatives have had their transfers cancelled.
“Amongst our clients are people who are disabled, who have survived torture and modern slavery and who have had traumatic experiences at sea. To house any human being in a ‘quasi floating prison’ like the Bibby Stockholm is inhumane. To try and do so with this group of people is unbelievably cruel.
"Even just receiving the notices is causing them a great deal of anxiety."
He said people should be housed in communities, not barges.
"The government could just get on with processing people’s asylum claims, instead they are playing to a gallery that seems to thrive on human suffering. We will continue supporting people to challenge their decision," said Mr Smith.
Cheryl Avery, the Home Office’s director for asylum accommodation, said 15 people have been moved on to the vessel so far.
However, Care4Calais said about 20 asylum seekers did not board the barge as planned because their transfers were “cancelled” after lawyers challenged the decisions.
Last month, a whistleblower in the local authority told The Times that checks of the vessel had raised “serious” concerns, which could potentially lead to a fire similar to that which destroyed Grenfell Tower in North Kensington, London, killing 72 people in 2017.
Meanwhile, the government is preparing to increase fines given to employers and landlords who allow people arriving by irregular means to work for them or live in their properties.
It is reportedly also reconsidering flying people who arrive by unauthorised means to Ascension Island if the scheme to send them to Rwanda fails.
Immigration Minister Robert Jenrick insisted the barge is a “safe facility” after the Fire Brigades Union called it a “potential deathtrap”, citing concerns such as overcrowding and a lack of access to fire exits.
“We hope that the first migrants will go on to the boat in the coming days, I'm not going to give you an exact date – but very soon,” Mr Jenrick told Sky News on Monday.
He reiterated the government's plan to house 500 people on the barge, despite concerns from the firefighters' union over the vessel having initially been designed to house only about 200.
However, Number 10 appeared to suggest a minister had misspoken when she said it was hoped the Bibby Stockholm barge could be up to its full 500 capacity by the end of the week.
The Prime Minister’s official spokesman, asked about safeguarding minister Sarah Dines’ comment, said: “My understanding, and I think the Home Office have sought to clarify, that is about the upward capacity of the Bibby Stockholm rather than the numbers we are looking to get in by the end of the week.
Asked whether the minister misspoke, the spokesman replied: “All I would say is that my understanding is that the Bibby Stockholm has an upward capacity of 500.“We are looking to (reach) that number over time — I don’t think we are aiming to do it by the weekend.”
Meanwhile, Labour has accused the Government of “disastrous failure” over its pledge to stop the boats after Home Office figures released on Monday showed the number of migrants staying in hotels has hit 50,000.
Shadow home secretary Yvette Cooper said the figure represented a 25 per cent increase from 40,000 in December, when Rishi Sunak promised to end the placement of asylum seekers in hotels.
The developments come during the government's “small boats week”, in which it is making a series of announcements on the issue, which Prime Minister Rishi Sunak has promised to solve.
Inside the Bibby Stockholm barge - in pictures
Civil penalties for employers will be increased up to a maximum of £45,000 ($57,350) per worker for a first breach and £60,000 for repeat offenders, tripling both from their current level, which was established in 2014.
Landlords face fines of £1,000 per occupier to £10,000, with repeat breaches going from £3,000 to £20,000. Penalties relating to lodgers will also be raised.
Landlords and employers are required to check the eligibility of their employees and tenants.
“Making it harder for illegal migrants to work and operate in the UK is vital to deterring dangerous, unnecessary small boat crossings,” Mr Jenrick said.
Mr Sunak, who became Prime Minister in October, has pledged to stop thousands of asylum seekers crossing the English Channel following a sharp rise in arrivals.
Last month his government passed a law, criticised by the UN, that bars asylum claims by migrants arriving via the Channel and other “illegal” routes.
It also mandates their transfer to third countries, such as Rwanda, but that element of the law has been bogged down by court challenges.
If that plan fails, the government is said to be considering flying people who arrive in the UK by unauthorised means to Ascension Island.
The tiny volcanic outcrop in the South Atlantic could host an asylum-processing centre in an attempt to reduce the number of small boats crossing the Channel under the scheme.
Where is Ascension Island?
Ascension Island is a British Overseas Territory, about 6,400km from the UK in the South Atlantic Ocean, east of Brazil and west of Angola.
The island is 1,300km north-west of St Helena and about 88 square kilometres in size.
Who lives on Ascension Island?
There is no permanent or indigenous population. Rather, it is inhabited mostly by St Helenians, Americans and Britons who work on the island, and their families.
About 800 people live on the island.
What amenities are on the island?
The island has two settlements, Georgetown and Two Boats.
It has two military bases, which are home to Britain's Royal Air Force and the US Air Force, along with an airfield and a BBC relay station.
Ascension Island has a handful of shops, gyms and swimming pools, and even a cinema and an array of outdoor activities.
Who governs the island?
Ascension Island is not a part of the UK, it is self-governing, with its own constitution that it shares with St Helena and Tristan da Cunha, also British Overseas Territories in the South Atlantic. The UK is responsible for defence, international relations and security on the island.
Ascension Island is administered by five councillors and the governor of St Helena, Nigel Phillips.
The currency on the island is the St Helena pound, which has the same value as pound sterling.
The island was used as a base by the US in the Second World War and as a support base for the RAF in the Falklands War.
What is the environment like on the island?
The temperature ranges from 20ºC to 38ºC with a wet season from January to April.
The Green Mountain has a summit at 859 metres but the island also has picturesque coastal areas and an array of native wildlife, including turtles, sea birds and crustaceans.