Doctors asked Italian authorities to allow 16 people ashore on Saturday for urgent medical treatment after they had spent nearly a week trapped on a small rescue boat docked in Sicily while officials squabbled about their final destination.

Another 134 refugees and migrants last night remained on board the coastguard vessel, which is less than 100 metres long.

On Friday they launched a hunger strike in protest at their situation. Almost all on board come from Eritrea, a country with one of the world’s worst human rights records, and so are likely to have a strong claim for asylum.

The UN has urged European countries to “do the right thing” and take in some of the group, after Italy said it would not let anyone disembark until other EU member states agreed to offer them homes.

“The time has come to end the back-and-forth that has seen countries competing in a race to the bottom on who can take the least responsibility for people rescued at sea,” the UN’s high commissioner for refugees, Filippo Grandi, said in a statement on Saturday. “I urge European states to uphold [their] principles, to do the right thing and offer places of asylum for people rescued from the Mediterranean Sea in their time of need.”

The crisis over the people effectively held prisoner in the Sicilian port of Catania on board the Diciotti has lasted almost a week. The Diciotti rescued 190 people from an overcrowded boat off the Italian island of Lampedusa on 15 August. Thirteen were evacuated for emergency medical treatment, and after the boat was turned away by Maltese authorities it was allowed to dock in Sicily.

But Matteo Salvini, Italy’s far-right interior minister, said he would not allow those on board to disembark until he was assured all would “go elsewhere”.

He eventually allowed 27 unaccompanied children off the boat on Wednesday, and on Saturday doctors identified those who needed medical care. Two of the 16 had possible cases of tuberculosis.

Italian prosecutors in Sicily have already launched an investigation against “unknown people”, looking at the possible illegal detention of those on board. They have sent a team from the regional city of Agrigento to Rome to question officials.

But Salvini hit back at the local officials in a broadcast on Facebook Live. “I heard that the prosecutor’s office in Agrigento has opened an investigation,” he said. “I also heard that the suspects are ‘unknown’ at the moment. But I’m not unknown. My name is Matteo Salvini, I’m the minister of the interior. Come on, try me too, I’m here.”

The deputy prime minister has also hit out against critics in Europe, threatening to suspend the country’s financial contribution to the EU if Brussels did not intervene to redistribute the people on board. The European commission called the threats unhelpful.