It could take Britain 10 years to negotiate a trade deal with the US if it votes to leave the EU, Barack Obama has said.

Speaking to the BBC at the end of his final visit to the UK as president, Obama said: “It could be five years from now, 10 years from now before we’re actually able to get something done.”

Obama is in the last nine months of his presidential term and has spent the past three days in London as part of a state visit. During that time he urged Britons to remain part of the EU as they prepare to vote on membership of the 28-country bloc at the 23 June referendum.

Obama told the BBC that Britain would not get preferential treatment over the EU when it came to negotiating a new trade deal. He said: “The UK would not be able to negotiate something with the United States faster than the EU. We wouldn’t abandon our efforts to negotiate a trade deal with our largest trading partner, the European market.”

The president added that, despite criticism of him, he thought ordinary British voters would be interested in his thoughts on the UK and the EU. “Since we rely heavily on the UK as a partner globally on a whole range of issues, we like you having more influence, we like you being at the table helping to influence other countries who may not often see things as clearly – from our perspective – as our British partners do,” he said.

“I don’t anticipate that anything I’ve said will change the position of those who are leading the campaigns in one direction or the other, but for ordinary voters I thought it would be relevant to hear what the president of the United States, who loves the British people and cares deeply about this relationship, has to say.”

Obama’s comments came as US presidential hopeful Hillary Clinton also threw her weight behind the campaign to keep Britain in the EU in a major new boost to David Cameron’s remain campaign.

Clinton’s senior policy adviser Jake Sullivan told the Observer: “Hillary Clinton believes that transatlantic co-operation is essential, and that co-operation is strongest when Europe is united. She values a strong British voice in the EU.”

US intervention in the EU debate has angered leave campaigners, who have repeatedly argued that Britain could easily negotiate deals and get better terms outside the EU.

But a Downing Street source said: “When you face such a big decision in life, most people listen to their friends, and we disregard such advice at our peril.”

The home secretary, Theresa May, rejected the idea that Obama was lecturing British voters, insisting he was simply setting out the facts.

“What he said was, lots of people are talking about the sort of trade deal that the UK could have with the US if it was outside the EU. As president of the United States, he said he would tell us what his view was, how he saw the facts from his point of view,” she told BBC1’s The Andrew Marr Show on Sunday.

May was once regarded by some in the leave campaign as a potential figurehead; but she said her experience of tackling immigration, security and other issues as home secretary had convinced her that the UK would be better off if it remained an EU member.

“I think we have a positive, optimistic future inside the European Union,” she said. “I think our prosperity looks more secure.

“Looking across the issues of security, trade and the economy, all of those say to me that we will be better off inside the EU, and I think people will listen to our friends and allies, will want to hear the facts.”

She conceded that managing migration to the UK would continue to be a tough challenge if Britain remained inside the EU, but repeatedly sought to dismiss the idea that accepting the principle of free movement – which allows EU citizens to live and work in the UK – meant being unable to control the borders. “We have checks at our borders, we have controls at our borders,” she said.

Obama’s comments drew an angry response from the mayor of London, Boris Johnson, who came under fire last week for saying Obama’s “part-Kenyan” heritage had driven him towards anti-British sentiment.

Johnson launched a further attack on the president’s “ridiculous and weird” arguments for Britain to stay in the EU. He told the Mail on Sunday that Obama was entitled to his view but that “it is ridiculous to warn that the UK will be at the back of the queue for a free trade deal”.

Johnson said the only reason the UK had not already got a trade deal with the US was because being a member of the EU hampered negotiations.

In a separate interview, the mayor was asked if he should apologise for his comments about Obama, to which he responded: “Oh come on. This is all a complete distraction – an attempt by the remain campaign to throw dust in people’s eyes.”

He said it was “inconsistent, perverse and, yes, hypocritical” to tell the EU to give up its sovereignty when the US would not dream of doing a similar thing itself.

The former Tory defence secretary Liam Fox said on Friday night that Obama’s opinions would be irrelevant after the US presidential elections in November.

“Whoever it is that will be at the helm of the United States won’t be Barack Obama,” Fox told BBC2’s Newsnight. “It will be the next president, and the next Congress, who will be in charge of any trade arrangements.”

During the BBC interview, Obama also said that it would be a mistake for any western state to send ground troops into Syria, and that countries could not “pull up the drawbridge” in the migration crisis.

On Saturday, the president addressed an audience of 500 people, many aged between 18 and 30, at a town hall-style event in central London, during which he said the UK’s role in the EU had helped secure peace on the continent.

Obama urged the young audience to reject isolationism and xenophobia and to “take a longer and more optimistic view of history”.