David Cameron has undermined one of Nick Clegg's flagship policies for improving social mobility, saying it is "fine" to offer his friends' children internships and even admitting that he has given a work placement to a neighbour.

The government has put more accessible internships in desirable professions at the centre of a drive to give poorer children better opportunities. Earlier this month Clegg, the deputy prime minister, admitted securing a "definite leg-up internship" through his father's influence in a Finnish bank. He said it was wrong that his career had been boosted by parental connections.

But in an interview in the Daily Telegraph, Cameron said he was "very relaxed" about offering work placements to people he knew. "I've got my neighbour coming in for an internship," he said. "In the modern world, of course you're always going to have internships and interns – people who come and help in your office who come through all sorts of contacts, friendly, political, whatever."

Earlier this month, Clegg, the deputy prime minister, told the Commons : "As a teenager, yes, I did receive an internship, as, I suspect, did many people around the chamber. Good for you if you did not. All of us should be honest and acknowledge that the way that internships have been administered in the private sector, the public sector, political parties and – I discovered when we came into government – in Whitehall as well, under 13 years of Labour, left a lot to be desired."

Clegg later claimed professional life should be "about what you know, not who you know". He said: "The whole system was wrong. I'm not the slightest bit ashamed of saying that we all inhabited a system which was wrong."

The revelation that the deputy prime minister was helped through his father's connections cast a shadow over the government's announcement of the drive to end unpaid internships.

Cameron said Clegg was "trying to make a fair point", but happily admitted that as a young man he, like his deputy, was helped out by his family connections.

The prime minister, who this week was also caught in a row over whether he will wear a morning suit at next week's royal wedding, denied he was trying to rewrite his background.

"People know who I am," he said. "I'm not trying to rewrite my background. I went to a fantastic school, I adored my parents."

But he added: "I suppose when I got into politics I was always called the Old Etonian David Cameron. "

In the Telegraph interview Cameron also spoke about a recent visit he made to the grave of his son, Ivan, who died in 2009. He said: "The first person who says to you, 'Soon you'll think of the happy memories of him and you won't be so sad' … well, you want to deck them. But actually, it is true that, suddenly, some happy memories burst through the cloud."

Cameron also likened welcoming Lady Thatcher to No 10 as an "out of body experience".

• This article was amended on 23 April 2011. The original referred to the offering of internships to children's friends. This has been amended.