The EU says it will pay €2,000 (£1,770; $2,225) each to migrants 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 Thursday. It was agreed with the Greek government.
She said it was temporary – open for one month only – and only for migrants who arrived before 1 January.
She said 5,000 migrants would be eligible for the “voluntary return”.
This month, hundreds of migrants and refugees have reached Greek islands near Turkey by boat, increasing the pressure on struggling reception centres. 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 islands, says more than 14,000 of the migrants are children.
“Men, women and children are living in horrific conditions in these overcrowded centres, in constant fear and with very basic access to services like toilets, showers, electricity,” Stephan Oberreit, MSF head of mission in Greece, told the BBC.
“Our teams in the clinic opposite Moria camp receive around 70 children per day, including children suffering from chronic illnesses, for which we are not able to provide proper care.”
Ms 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 migrants 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.