On Wednesday, European Commission Vice President Valdis Dombrovskis told a news conference in Brussels that the "draft report concludes that Greece seriously neglected its obligations" under the Schengen agreement.
He said there were "serious deficiencies in the carrying out of external border controls that must be overcome and dealt with by the Greek authorities".
Spot checks by EU inspectors in November found Greece was failing to register arrivals properly, to fingerprint everyone, to see whether identity documents were genuine, and to check people against Interpol and other databases.
EU Commissioner for Migration Dimitris Avramopoulos conceded that Greece had "started undertaking efforts towards rectifying and complying with the Schengen rules" since then.
However, "substantial improvements" were needed.
Responding to the draft report, the Greek government said it was "not constructive to isolate Greece for creating a media impression", based on outdated information.
A spokeswoman said Greece had repeatedly asked for help from the EU's border force Frontex to boost controls, but that the response had been "small".
Criticising a lack of EU progress on the crisis, she said: "Greece has done more than it could to stand to its obligations. We expect from everyone else to do the same."
The report must be approved by other Schengen members before the Commission can set the three-month deadline.
But this is a serious warning to Greece to improve things, says the BBC's Damian Grammaticas in Brussels.
Otherwise, border checks to limit the movement of migrants within Europe may be instituted, preventing those in Greece from travelling north.
Timing is crucial - by the BBC's Damian Grammaticas in Brussels
The prospect of many thousands more refugees and migrants coming this summer is what's worrying the countries they head for, primarily Austria, Germany and Sweden.
Germany's Chancellor Angela Merkel told an audience on Tuesday: "Of course we must significantly reduce the number of refugees and we are working on that."
So it's likely Greece will now be given until mid-May to get to grips with the influx of people to its islands.
And the timing is crucial. Mid-May could be when a summer surge of refugees reaching the EU is starting to peak.
The temporary border controls which countries like Germany have in place will be due to expire.
To prolong the controls governments will need an official assessment by the European Commission that Greece is still failing.
He also told BBC Newsnight that Belgium had told his country to "push" migrants "back in the sea" as a solution to the crisis.
However, Belgian Migration Minister Theo Francken condemned the claims, calling them "grotesque and very regrettable".
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.