Net migration to the UK rose to 333,000 in 2015, according to Office for National Statistics estimates - the second highest figure on record.
Net migration is the difference between the number of people coming to the UK for at least a year and those leaving.
The figure for EU-only net migration was 184,000, equalling its record high.
Boris Johnson said David Cameron had been "cynical" to promise to bring net migration down to below 100,000 while the UK was part of the EU.
Speaking to BBC Political Editor Laura Kuenssberg, the Conservative MP and former London Mayor said it was "cynical and unacceptable to say you can fulfil that pledge".
"I think that they (the figures) show the scandal of the promise made by politicians repeatedly that they could cut immigration to the tens of thousands and then to throw their hands up in the air and say there's nothing we can do because Brussels has taken away our control of immigration," he said.
Mr Johnson, a leading figure in the campaign to get Britain out of the EU in June's referendum, said he was pro-immigration but there was "no public consent for the scale of immigration we are seeing" and the situation was "completely out of control".
He said that the only solution was to leave the EU, saying that a vote to stay in the union would mean people "kissing goodbye permanently to control of immigration".
Laura Kuenssberg said the immigration figures "could hardly have come at a more sensitive time, just when the referendum campaign is moving into a more brutal and frantic phase".
Home Office minister James Brokenshire said David Cameron's renegotiation of the UK's membership of the EU, which will see restrictions placed on the benefits new arrivals can claim and a crackdown on "sham marriages", would "close back-door routes" into the country.
"Leaving the EU is absolutely no panacea or silver bullet," he added, telling BBC News that net migration from outside the EU was higher than from within it and leaving would "wreck the economy and harm jobs".
In other developments, with four weeks to go before the UK votes on 23 June on whether to stay in or leave the EU:
- The BBC will stage its first EU referendum debate in Glasgow later
- Alex Salmond and Liam Fox will be among MPs questioned by voters aged 18 to 29
- The Leave campaign says "foolish" EU procurement rules cost the UK £1.6bn a year
- The Scottish Parliament is to formally back the case for staying in the EU
- Ryanair is accused of breaking electoral law in its campaign for staying in the EU
- Nick Robinson looks at the issue of Indian restaurant chefs and the EU vote
According to the Office for National Statistics, there was a 20,000 rise in net migration to the UK from the 313,000 for the year to December 2014.
The figure is well above the government's aim of getting it to the "tens of thousands" and is the highest recorded for a calendar year.
A Downing Street spokesman declined to say whether Mr Cameron remained committed to that target but said the government would "continue to look at what more might be done".
The spokesman added: "Of course people rightly have concerns on immigration but the PM's view is very clearly that wrecking the economy and destroying jobs by getting rid of our privileged access to the world's biggest market is not the answer."
He said the Leave campaign failed to spell out what would happen to migration if the UK left the EU, with some saying "they will cut migration while telling other groups that they will keep freedom of movement or increase visas".
Thursday's figures show estimated levels of long-term migration into the UK from within the EU and outside in the year to December 2015.
They show that 270,000 EU citizens moved to the UK for at least a year in 2015, up from 264,000 in 2014. The number of non-EU citizens moving to the UK was 277,000, down from 287,000 in 2014.
The rise in the net figure was the result of a fall in the numbers of people emigrating.
Analysis By Home affairs correspondent Dominic Casciani
The net migration figure is just short of the record but that statistic alone doesn't do justice to the global picture and the UK's place in it. Everywhere you look, people are on the move - coming and going depending on economic and political circumstances.
EU immigration is relatively stable - although there are more southern Europeans than before thanks to the Eurozone's economic doldrums.
One really interesting figure, that doesn't affect the population headcount - officially at least - is the 110,000 rise in short-term immigration to about 1.2m.
More people are coming for less than a year - such as short contract workers selected by employers who are looking internationally for the best people at the most competitive price.
But it will also include some Brits who spend part of the year here - and the rest living in the sun.
UKIP leader Nigel Farage said the numbers show "mass immigration is still hopelessly out of control and set to get worse if we Remain inside the EU, going on with disastrous open borders".
But former Labour Home Secretary David Blunkett warned that an agreement with France to allow British border controls at Calais and the Eurostar terminals in Paris and Brussels would be at risk if Britain left the EU.
He criticised Leave campaigners for wanting to create a "fortress Britain" - and defended the last Labour government's record immigration, saying "We didn't just simply say to Poles coming in back in 2004, just come here, we said we'd rather you come here and work than be in the sub-economy."
He added: "40% of those who registered to work back in 2004 were already in the country and the reason I make that point is because a lot of people would come in illegally if we didn't have the border controls on French soil."
The Institute of Directors said the public was "being very poorly served by the current level of the debate on immigration" and people came to work in the UK because it is one of "best performing economies in Europe".
"Employment is at a record high, and there are big skills gaps that employers cannot fill domestically. Despite the overblown rhetoric that today's figures have sparked, the actual numbers have not changed much in the last year," said IoD employment spokesman Seamus Nevin.
Other ONS figures on employment show:
- The estimated employment level of EU nationals (excluding British) living in the UK was 2.1 million in January to March 2016 - 224,000 higher than the same quarter last year, according to the Labour Force Survey
- British nationals in employment increased by 185,000 to 28.2 million and non-EU nationals in employment increased by 5,000 to 1.2 million
- Over half of the growth in employment over the last year was accounted for by foreign nationals
- There were 630,000 National Insurance number registrations by EU nationals in the year to March 2016, an increase of 1,000 on the previous year. For non-EU nationals, there were 195,000 registrations, an increase of 2,000 on the previous year.
How to follow the BBC One EU referendum debate
- The first of three BBC referendum debates, hosted by Victoria Derbyshire, will be broadcast on BBC One from 20:00 BST
- There will also be live coverage on the News Channel, 5 Live and Newsbeat
- You can follow the action minute-by-minute on the BBC's Live referendum page
- The BBC News Facebook page will livestream before and after the debate
- The BBC News Snapchat Channel will also be active before, during and after the debates