The German-flagged ship had rescued 40 refugees from a sinking vessel off the coast of Libya. Italy's Interior Minister Matteo Salvini has accused Germany of "blackmail" in the refugee resettlement issue.
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German NGO Sea-Eye on Thursday said it plans to approach the Italian island of Lampedusa but remain outside Italian territorial waters, after Italy's hard-line interior minister, Matteo Salvini, banned the migrant rescue vessels.
The Alan Kurdi ship rescued 40 migrants on Wednesday night, including a baby, two small children and two pregnant women.
The migrants said they had set off from the Libyan city of Tadschura, east of the capital, Tripoli.
Sea-Eye spokesperson Carlotta Weibl said migrant rescue boats had to "respond immediately to situations, as soon as there is an emergency at sea."
"Political calculations should not play any role whatsoever," she said. The closest port is at the Italian island of Lampedusa, and Weibl said it was critical that migrants be allowed to disembark there in accordance with international law so that they could receive future treatment.
Weibl said the ship had tried to make contact with the Italian coast guard, but has yet to receive a response.
On Thursday, Salvini accused Germany of telling the European Commission that it would not accept a group of migrants from the Gregoretti unless Italy accepted those rescued on the Alan Kurdi.
"This is real blackmail," said Salvini. "It confirms that other European countries consider Italy as their refugee camp, but things have changed and we no longer accept orders and invasions."
There was no comment from Berlin about the accusation.
The EU's Dublin III Regulation says migrants are to apply for asylum in the first EU country they enter the bloc. Italy, Malta and Greece, as the countries closest to Libya, have received the most migrants, and have argued that they are unable to settle them all.
Italy says there could be criminal consequences if a ship docks in their ports. Authorities arrested migrant rescue ship captain Carola Rackete in July, after she docked in Lampedusa without prior permission. She has since been released following an international outcry.
How did Europe's refugee crisis start?
From escalating violence in the Middle East and Africa to incoherent asylum policy at home - DW looks at how the EU has found itself in the midst of a refugee crisis.
Image: picture-alliance/dpa
Fleeing war and poverty
In late 2014, with the war in Syria approaching its fourth year and Islamic State making gains in the north of the country, the exodus of Syrians intensified. At the same time, others were fleeing violence and poverty in countries such as Iraq, Afghanistan, Eritrea, Somalia, Niger and Kosovo.
Image: picture-alliance/dpa
Seeking refuge over the border
Vast numbers of Syrian refugees had been gathering in border-town camps in neighboring Turkey, Lebanon and Jordan since 2011. By 2015, with the camps full to bursting and residents often unable to find work or educate their children, more and more people decided to seek asylum further afield.
Image: picture-alliance/dpa
A long journey on foot
In 2015 an estimated 1.5 million people made their way on foot from Greece towards western Europe via the "Balkan route". The Schengen Agreement, which allows passport-free travel within much of the EU, was called into question as refugees headed towards the wealthier European nations.
Image: Getty Images/M. Cardy
Desperate sea crossings
Tens of thousands of refugees were also attempting the perilous journey across the Mediterranean on overcrowded boats. In April 2015, 800 people of various nationalities drowned when a boat traveling from Libya capsized off the Italian coast. This was to be just one of many similar tragedies - by the end of the year, nearly 4,000 refugees were reported to have died attempting the crossing.
Image: Reuters/D. Zammit Lupi
Pressure on the borders
Countries along the EU's external border struggled to cope with the sheer number of arrivals. Fences were erected in Hungary, Slovenia, Macedonia and Austria. Asylum laws were tightened and several Schengen area countries introduced temporary border controls.
Image: picture-alliance/epa/B. Mohai
Closing the open door
Critics of German Chancellor Angela Merkel's "open-door" refugee policy claimed it had made the situation worse by encouraging more people to embark on the dangerous journey to Europe. By September 2016, Germany had also introduced temporary checks on its border with Austria.
Image: Reuters/F. Bensch
Striking a deal with Turkey
In early 2016, the EU and Turkey signed an agreement under which refugees arriving in Greece could be sent back to Turkey. The deal has been criticized by human rights groups and came under new strain following a vote by the European Parliament in November to freeze talks on Turkey's potential accession to the EU.
Image: Getty Images/AFP/A. Altan
No end in sight
With anti-immigration sentiment in Europe growing, governments are still struggling to reach a consensus on how to handle the continuing refugee crisis. Attempts to introduce quotas for the distribution of refugees among EU member states have largely failed. Conflicts in the Middle East and elsewhere show no signs coming to an end, and the death toll from refugee sea crossings is on the rise.