Thousands of refugees have arrived on Europe's shores since the beginning of the year, with many drowning while attempting to cross the Mediterranean.
As EU member states - particularly Greece and Italy - struggle to cope with the migrant influx, parliament's Socialist and Liberal group leaders have reiterated the need to find a common European solution to what Amnesty International has dubbed "the worst refugee crisis since the second world war".
Earlier this year at June's council meeting, EU heads of state agreed to a "temporary and exceptional" relocation scheme for 40,000 asylum seekers entering Italy and Greece spread out over two years.
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Member states were given quotas for how many migrants to welcome, however these are not binding, with for example the UK opting out of the plan.
Parliament's Group of the Progressive Alliance of Socialists and Democrats leader Gianni Pittella called the outcome of that summit "farcical" and accused member states of having "contributed to the ill feeling growing in Europe - helping build walls rather than bridges between borders and people".
The Italian deputy called for "a complete overhaul of the [so-called] Dublin agreement, replacing it with a real and effective common European policy based on solidarity".
The Dublin regulation determines which member state is responsible for processing an asylum claim, usually the country where the refugee first entered the EU.
"German chancellor Angela Merkel and French president François Hollande's call for frontline countries to immediately build new registration centres is understandable but only on the condition that the whole of Europe puts aside national selfishness and comes together to draw up a common mandatory system capable of dealing with this humanitarian crisis in the short and mid-term", he added.
Pittella urged the EU to "call for an international summit involving the main actors such as the UN, Turkey, Russia and responsible north African countries to find a credible lasting solution".
Alliance of Liberals and Democrats for Europe group leader Guy Verhofstadt stressed the need for "a completely new European approach, and this can only be achieved if all 28 member states sit at the table, together with the president of the European council".
"The refugee crisis needs a long-term European solution and it has to be agreed under the leadership of the European commission and the European council. Only under this umbrella will we be able to replace the Dublin regulation with a centralised European asylum system that allocates refugees more fairly between EU countries".
MEPs are set to discuss measures to assist Greece and Italy at September's plenary session, with a debate currently scheduled for 9 September.