The "Out" campaign stood three points ahead of "In" in each of the two surveys for the Guardian newspaper, one of which was conducted online and the other by telephone.
They were conducted over three days to May 29 after official figures showed on May 26 that British net migration hit the second highest level on record last year. Last week, leaders of the Out camp turned their focus back on migration.
Britons will vote on June 23 on whether to remain in the 28-member EU, a choice with far-reaching consequences for politics, the economy, defense and diplomacy in Britain and far beyond.
From US President Barack Obama to the International Monetary Fund, a host of world leaders and international organizations have cautioned British voters about the risks of leaving the bloc it joined in 1973. The Bank of England has said a British exit, or Brexit, could tip the economy into recession.
Despite the warnings, Out has appeared to gain traction by focusing on the issue of migration. Many voters are concerned about the strains placed on schools, hospitals and housing from people moving to live in Britain.
ICM said the polls published on May 31 gave Out its first lead in one of its telephone surveys.
Telephone polls have previously tended to give the In campaign a comfortable lead and the sharp swing wrong-footed financial markets, sending sterling to a one-week low against the US dollar.
Polls conducted over the Internet have largely suggested a tighter race but ICM's latest weekly online poll - separate to its phone survey - showed voters favoring Britain leaving the EU by 47% to 44%.
