EU officials accuse Greece of refusing to accept help in accepting more refugees. Greece would become the first country to be suspended since passport-free travel was established in 1985.
More than 740,000 people have arrived on the shores of Greece in 2015 before moving north across Europe.
A top EU official has warned that children are dying on Greek beaches because of a lack of medical help.
There is also anger that Greece has been slow to let in EU experts and improve its registration of refugees.
The influx of refugees across countries with few border restrictions has alarmed European leaders and raised doubts about the very future of the 1985 Schengen accord.
According to the Financial Times, Greece has been repeatedly warned this week that it could be suspended from the 26-member zone if it fails to take action before an EU leaders’ summit in mid-December.
Separately, unnamed officials in Brussels have said that a suspension has been discussed in recent days in an attempt to persuade the Athens government to improve its handling of migrants and refugees.
Greece is not only struggling to cope with the numbers of people arriving on the islands closest to Turkey such as Lesbos, but thousands of people are camped on the northern border with Macedonia, which has erected a fence to stop people crossing.
Greece’s EU partners are frustrated that the Athens government has been slow to accept offers of help from the border agency, Frontex, to better manage its sea border. The offer is particularly sensitive in Greece because of concerns over its national sovereignty.
While suspension from the Schengen accord would directly affect Greeks travelling to the 25 other members of the passport-free zone, it is unclear what effect it would have on the refugee crisis.
None of the Balkan countries on the refugee route between Greece and Hungary are part of Schengen.
Greek businesses are becoming increasingly concerned by the crisis on the Macedonian border because protests on the railway line have halted rail traffic.
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