The EU will pay €2,000 ($2,225) each to refugees in overcrowded camps on the Greek islands willing to go back to their home countries.
EU Home Affairs Commissioner Ylva Johansson announced the scheme in Athens on March 12.
The scheme was agreed with the Greek government.
Ylva Johansson said it was temporary – open for one month only – and only for refugees who arrived before January 1.
The commissioner said 5,000 migrants would be eligible for the “voluntary return”.
This month, hundreds of immigrants and refugees have reached Greek islands near Turkey by boat, increasing the pressure on struggling reception centers. The camps on those islands already have nearly 42,000 asylum seekers, though they were designed for about 6,000.
Aid agency Medecins Sans Frontieres (MSF), which is working on the Greek islands, says more than 14,000 of the refugees are children.
Ylva Johansson said seven EU member states had agreed to take in at least 1,600 unaccompanied children from the camps, seen as especially vulnerable.
Many of the refugees are Syrians fleeing the civil war, but there are also Afghans, Pakistanis and West Africans. It is not clear how many would qualify for refugee status.
Aid agencies consider Syria too dangerous for refugees to be sent back there, but some other countries of origin, such as Pakistan, are considered safe enough.
Greece has temporarily suspended its processing of new asylum applications – a move condemned by aid groups.
The latest surge in numbers at the Greek border came after Turkey announced that it would no longer stop them trying to enter Greece. Turkey, which is hosting 3.7 million Syrian refugees already, accuses the EU of not doing enough to help.
Ylva Johansson said repatriation of refugees from the islands would be coordinated with the UN’s International Organization for Migration (IOM) and the EU border force Frontex.
The situation is also acute on the Greece-Turkey land border, where Greek police have used tear gas and water cannon to keep immigrants out.