The UK will face a future migration "free-for-all" unless it leaves the European Union, Justice Secretary Michael Gove has warned.
Writing in the Times, he said any future expansion of the EU would pose a "direct and serious threat" to public services in the UK, including the NHS.
Downing Street said the accession of more countries was years away.
Home Secretary Theresa May said the UK was able to control its borders by blocking entry to terrorists.
Three current and former cabinet ministers - Michael Gove, Iain Duncan Smith and Owen Paterson - backing EU exit are all focusing on immigration as the Leave campaign tries to wrestle back the initiative after days of headlines about the economic risks of EU exit following US President Barack Obama's intervention in the referendum debate.
Mr Gove warned that five potential new members of the EU - Macedonia, Montenegro, Serbia, Albania and Turkey - would result in millions more people having the right to move to the UK.
"Because we cannot control our borders - and because our deal sadly does nothing to change this fact - public services such as the NHS will face an unquantifiable strain as millions more become EU citizens," he wrote.
"There is a direct and serious threat to our public services, standard of living and ability to maintain social solidarity if we accept continued EU membership," he added.
EU membership discussions for Macedonia, Montenegro, Serbia, Albania and Turkey - which started accession talks in October 2005 - are ongoing.
Countries can join the EU if they meet the criteria, which includes democracy, the rule of law, a market economy and adherence to the EU's goals of political and economic union and in the final stages of membership a country's accession treaty has to be ratified by every individual EU country.
A Downing Street spokeswoman said the UK had a veto over other countries joining and said David Cameron had made clear there would need to be a different set of "transitional arrangements" in future.
Giving her first major speech of the EU referendum campaign, Mrs May said free movement rules did make it harder to control the volume of immigration, adding: "But they do not mean we cannot control the border."
She said the response in the wake of the Paris and Brussels attacks "cannot be to say that we should have less cooperation with countries that are not only our allies but our nearest neighbours", adding: "My judgment is that remaining a member of the European Union means we will be more secure from crime and terrorism."
Earlier former Work and Pensions Secretary Iain Duncan Smith, another leading Leave campaigner, said immigration was "out of control" and free movement of people rules meant the UK had to accept criminals coming into the country.
"An elected government in the UK elected on a platform to reduce immigration right now cannot deliver that because the EU is an open border," he told Radio 4's Today.
"You cannot reject anyone unless you can demonstrate categorically that they pose an immediate threat to the life and livelihood of the UK."
Pro-Brexit campaigners have attempted to return the EU debate to the issue of immigration after President Obama warned that the UK would be at the "back of the queue" for trade deals with the US, if it voted to leave the EU on 23 June.
His intervention has been seen as a major boost for the Remain campaign.
But Mr Duncan Smith suggested Mr Cameron and US President had had a "cosy conversation" in advance about what Mr Obama would say.
The US was driven by national self-interest, he said, and "the British people would be picking up the pieces not the Americans".
Boris Johnson said the EU had vetoed "modest" restrictions on migration, such as requiring anyone coming to the UK to work to have a firm job offer.
"The home secretary formally proposed only if you have a job offer," the Mayor of London said.
"That to me seemed reasonable. That vanished. They said 'bog off'. The point I'm trying to make now...if you want to imagine what the future is like in the EU look at that deal and the poverty of what was achieved."
And in a speech later, former environment secretary Owen Paterson will say that the UK risks becoming "a colony on the edge of the European empire" if it votes to remain.