Immigration is harder to control due to the EU's free movement of labour rules, Home Secretary Theresa May has admitted.
However, in her first major intervention in the Brexit battle, Ms May insisted the task was not impossible as she tried to distance herself from an official Treasury study showing that immigration would soar by three million by 2030.
The move came as US president Barack Obama again waded into the Brexit battle, warning the UK would have to wait up to a decade for a trade deal with America if it quits the EU.
Unbowed by a furious backlash from the Leave camp against "interference" in British affairs, Mr Obama moved to explain his stark statement that the UK would be at "the back of the queue" for a beneficial economic arrangement if it breaks away from Brussels.
Insisting he had not sought to "scare" Britons into rejecting Brexit, Mr Obama told the BBC: "My simple point is that it's hard to negotiate trade deals. It takes a long time.
"We wouldn't abandon our efforts to negotiate a trade deal with our largest trading partner, the European market, but rather it could be five years from now, 10 years from now, before we were able to actually get something done."
The Home Secretary's remarks on immigration were seized on by the Leave camp as proof Britain had an "open door" policy.
Ms May told the BBC's Andrew Marr Show: "Yes, free movement makes it harder to control immigration, but it does not make it impossible to control immigration."
The Cabinet heavyweight. who has kept a low profile in the referendum debate since backing Prime Minister David Cameron's Remain stance, would not be drawn on a Government document saying immigration would jump by three million.
"That was an independent figure, that was an independent estimate," the Home Secretary said of the Office for National Statistics' prediction of a steep rise in immigration, which was put out by the Treasury as part of the Government's claim that Brexit would cost each household £4,300 a year.
The Home Secretary acknowledged migrants would look at the rises in the national living wage when making a decision on whether to come to Britain.
"I think, yes, obviously, the national living wage is going up and people look when they are going to move at what they are going to earn," Ms May said.
The Home Secretary insisted Britain had won a landmark concession from Brussels on immigration in that the EU will overturn judgments of the European Court of Justice that make it is easier for people to abuse the free movement rules.
In an explosive escalation of the highly personalised attacks launched against Mr Obama by the Leave campaign, Ukip leader Nigel Farage said the president's priority in attacking Brexit was to "guarantee his financial future".
"There is a bigger motivation and that is, with this trip, and with what he has said, he has guaranteed his financial future. He has done the bidding of the giant American corporates who want Britain to stay in the EU," Mr Farage told Sky's Murnaghan.
Prominent Labour MP Chuka Umunna accused London mayor Boris Johnson of being "unfit" to be prime minister after his remarks drawing attention to Mr Obama's Kenyan heritage.
Brexit would make households £40 a week better off, one of Margaret Thatcher's economic gurus, Professor Patrick Minford, told The Sun on Sunday.