Migrants on the Greek islands are to be offered €2,000 (£1,764) per person to go home under a voluntary scheme launched by the European Union in a bid to ease desperate conditions in camps.

The amount is more than five times the usual sum offered to migrants to help them rebuild their lives in their country of origin, under voluntary returns programmes run by the United Nations’ International Organization for Migration.

The offer will last one month, as the commission fears an open-ended scheme would attract more migrants to Europe. It will not apply to refugees who have no homes to return to, but is intended to incentivise economic migrants to quit the islands.

The EU’s home affairs commissioner, Ylva Johansson, said the scheme was “a window of opportunity for a targeted group”, adding that the IOM would run the scheme with the EU border agency Frontex.

“Refugees will not return, of course, they can’t return, but economic migrants that maybe know they will not get a positive asylum decision could be interested in doing that,” she told a small group of reporters.

The scheme, she said, could be a quick way to relieve the pressure on camps on the Greek islands, where conditions are “totally unacceptable”.

The commission hopes 5,000 people will take up the offer, although it acknowledged it lacks statistics on how many people on the Greek islands were economic migrants, rather than refugees. Migrants on the Greek mainland are likely to be offered extra money to leave – much less than €2,000, but higher than the usual resettlement sum of €370.

Since 2016, 18,151 people have chosen to return to their country of origin from Greece under a voluntary returns programme, funded by the EU and run by the IOM. Only about one-fifth (3,927) of returnees were on the islands.

The extra money will come from the €700m emergency aid for Greece from the EU budget promised by the commission earlier this month, after Turkey announced it was “opening the gates” and sending migrants to Europe.

Even before the recent flare-up in tensions at the Greek-Turkish border, the UN refugee agency reported last month that more than 36,000 asylum seekers were staying in reception centres across five islands, originally designed for 5,400 people.

Thousands are living in small tents with no electricity, heating or hot water. Health problems are rising, but shortages of medical staff mean residents struggle to get help.

More than 20,000 people were living at the Moria camp on Lesbos, up from 5,000 last July. About 85% of last year’s arrivals were refugees, with most coming from Afghanistan and Syria, but also from Iraq, Palestine, Somalia, the Democratic Republic of the Congo, and elsewhere.

More than 18,300 Moria residents were squashed into a facility designed for 2,200 while others were living in nearby olive groves.

Earlier this month doctors volunteering at Moria were forced to flee after being set upon by a mob carrying large sticks studded with nails. A series of attacks on aid workers has prompted some NGOs to withdraw from Lesbos, heightening concerns about the pressure on the Greek islands.