Taking in more refugees from Syria and other war-torn countries could help David Cameron secure European Union reforms, Labour leadership candidate Andy Burnham suggested.
Mr Burnham said Britain was failing to "play our part" in dealing with a dramatic rise in numbers seeking a safe haven in Europe after the discovery of 71 bodies in a truck in Austria heightened criticism of the continent's response to people smuggling.
He said that doing so could make other countries more likely to agree to wider changes around benefits and housing being sought by the Prime Minister as part of his renegotiation of EU membership.
Mr Cameron, criticised recently for describing those seeking refuge as a "swarm", faces competing political pressures to bring down record levels of net immigration into Britain and to help deal with the refugee crisis.
As Europe struggles to agree a co-ordinated response, his Government claims many are economic migrants and argues that the priority should be tackling the reasons people are leaving their own countries and closing down the people smuggling operations.
Mr Burnham told the Telegraph: "Germany and Scandinavia are taking far more than we are.
"We can't just say it is everyone else's problem and not ours, we have got to play our part. I think if we were to do that I think we might get more of what we want to negotiate on EU migration.
"If we want to have ideas about protecting wages or migrants accessing social housing, more financial support for communities affected by migration. If we play our part responsibly on the migrant crisis then we would [be more likely to get those things]."
No-one was "taking ownership of the problem", he complained.
"It is wrong to let Greece and Italy shoulder an enormous burden. I think there is a need for a coordinated European policy on immigration."
Labour leadership frontrunner Jeremy Corbyn said the Government needed to look at the issue "with renewed humanity" and stop treating migrants and those fleeing persecution "as criminals".
"These deaths are the consequences of desperate situations in many parts of the world, and the failure of those of us in more fortunate situations to do enough to create a more peaceful world," he said.
"These tragic deaths should remind us that we are all one human family."
Mr Burnham said it was vital for Labour to deal more directly with the issue of immigration, claiming Labour campaigners have been "avoiding people's eyes when this subject comes up on the doorstep".
He is spending the bank holiday weekend touring northern towns and cities with a pledge to win back more than a million voters lost to Ukip at the general election in May.
Extra EU funds for areas that face most pressure from new arrivals is among his proposed responses to a report commissioned from MP Dan Jarvis about the failure to take on Ukip.
Mr Jarvis said Labour was "in danger of becoming irrelevant" after falling to third place in some constituencies and was "yet to develop an effective counter attack" to the eurosceptic advance which saw it shed crucial support across a swathe of seats in its traditional northern heartlands.