EU countries would face international refugee quotas under proposals to be unveiled this week.
The suggestions are being put forward amid outcry over the number of migrants losing their lives as they try to cross the Mediterranean.
Nearly half of all asylum seekers in the European Union are currently accepted by Italy, Austria, Germany and Sweden.
Many of those who cross into Europe from Africa or from Turkey seek asylum as refugees, saying they come from countries affected by war or other crises.
The Times said all 28 member states will be obliged to make a "fair and balanced contribution" by taking in "persons in clear need of international protection".
It will involve Europe putting in place "a mandatory and automatically-triggered relocation system".
The Times said the number each EU country would be required to take in would be based on their GDP, population, unemployment and previous rates of taking in asylum seekers.
