Italy's Interior Minister Matteo Salvini made a partial concession to let 27 children stranded on board NGO migrant rescue ship Open Arms leave the vessel.
The minors were transferred on Saturday to a coast guard vessel and a border patrol boat for disembarkation and processing on the Sicilian island of Lampedusa.
Salvini said his climbdown was at the insistence of Prime Minister Giuseppe Conte and that the remaining migrants must stay on the ship.
It comes after the boat's captain, Oscar Camps, said the migrants were "psychologically broken" and the vessel was like a bomb about to explode.
He called for them to be evacuated immediately and said the NGO could no longer guarantee their safety.
The migrants, most of whom are African, were picked up off the coast of Libya and have been waiting to come ashore on the Italian island of Lampedusa for more than two weeks.
"After ... six medical evacuations and having told authorities about our situation without receiving an answer, we are in a situation of need and we cannot guarantee the security of the 134 people on board," Camps said.
"It is terrible, the things that are happening are not only physical but psychological. The conditions in which they were staying in Libya and now in the ship, it is just terrible, with 130 people and two toilets," he added.
France, Germany, Romania, Portugal, Spain and Luxembourg have said they will help relocate the migrants, but the reaction from Salvini's interior ministry has been sceptical.
Salvini issued a statement on Saturday reiterating that Open Arms could have taken the migrants to Spain and that it was to blame for their plight.
The president of the European Parliament, David Sassoli, has also been putting pressure on Italian authorities to do more.
"Today [Friday], my office was in contact with the captain of the Open Arms mission who described conditions on board as scarcely tolerable. The situation has become dramatic,” said Sassoli.
“The immigrants have been blocked on the vessel for already 14 days just one kilometre off the port of Lampedusa.”
“They are now giving up and are inflicting acts of self-harm, while they lose sense of reality.
"The hygiene conditions on board are worse than ever and it is necessary to allow immediate disembarkation of those on board.
"I hope that the Italian authorities understand the gravity of the humanitarian emergency on board the vessel and agree to let them enter the port today."
- Open Arms: Situation of migrants stranded at sea is 'untenable' says European Commission
- Open Arms captain says migrants are 'broken' and ship is like a ticking bomb
- Italian PM says six EU countries have offered to take the stranded Open Arms migrants