The European Commission will make the controversial proposal that EU member countries should take in refugees under a quota scheme.
The UN estimates that 60,000 people have already tried to cross the Mediterranean this year.
Migrants are being driven to make the journey by "horrific abuse" in Libya, according to Amnesty International.
More than 1,800 migrants have died this year in the Mediterranean, a 20-fold increase on the same period in 2014.
The European Commission's migration policy, to be announced on Wednesday, will also propose organising legal means for migrants to come to Europe so they do not turn to traffickers.
However, it will need to be agreed by EU states.
European leaders will discuss the proposals at a summit at the end of June.
'A crazy idea'
The measures proposed by the EC are the latest in a series of steps designed to stop migrants drowning in the Mediterranean.
More than 200,000 migrants fleeing conflict or poverty from countries such as Syria, Eritrea, Nigeria and Somalia are estimated to have crossed the Mediterranean last year, with thousands dying making the journey.
Quotas would be determined using a number of factors, including a country's population, economic indicators and the number of asylum seekers previously accepted.
Germany keenly supports the idea of quotas, having received 200,000 asylum applications last year.
Countries such as Italy and Malta, where large numbers of migrants arrive by boat, have also called for EU members to share responsibility for migrants more evenly.
Austrian Chancellor Werner Faymann has described a quota as "a question of fairness", adding that asylum is "not an act of mercy but a human right".
However, other EU countries are fiercely opposed to the idea of quotas.
A UK government spokesman said: "The UK has a proud history of offering asylum to those who need it most, but we do not believe that a mandatory system of resettlement is the answer."
"We will oppose any EU Commission proposals to introduce a non-voluntary quota."
Leaders in Hungary, Slovakia and Estonia have also objected to a quota system, with Hungarian Prime Minister Viktor Orban calling it "a crazy idea".
Mediterranean migrants: in numbers
- 219,000 people arrived in Europe
- 3,500 deaths/missing
In 2015 (1 Jan- 27 April):
- 46,000 arrivals in Europe
- More than 1,750 deaths/missing
On Monday the EU foreign policy chief, Federica Mogherini, will ask the UN Security Council to permit the use of force against people traffickers operating out of Libya.
Without a UN mandate, military action to destroy or halt traffickers' boats in Libyan or international waters would be illegal.
Libya's ambassador to the UN, Ibrahim Dabbashi, told the BBC that his country is against the proposal.
"The Libyan government has not been consulted by the European Union. They have left us in the dark about what their intentions are, what kind of military actions they are going to take in our territorial waters, so that is very worrying," he told the World Service's Newsday programme.
Amnesty International has warned that military action could leave migrants trapped in Libya in desperate conditions.
"Introducing measures to tackle smugglers without providing safe alternative routes out for the people desperate to flee conflict in Libya, will not resolve the plight of migrants and refugees," said Amnesty's Middle East and North Africa director Philip Luther.
The new Amnesty report is titled "Libya is full of cruelty". The title comes from a statement by an unnamed Nigerian migrant in the country.
Based on dozens of interviews of migrants the report documents stories of abduction, torture and rape.
It alleges that people smugglers are systematically abusing migrants. It also says that conditions in Libyan migrant detention centres are often appalling.
One Syrian family interviewed said that they were left with no choice but to attempt the perilous voyage.
"We were facing death in Libya so we thought we might as well face death in trying to get to Italy."
The report calls on the international community to "dramatically expand search and rescue operations".
European countries have already agreed to expand rescue services, having cut them last year.