A third of university graduates who are employed as interns receive no payment, while paying out up to £926 a month in living costs, according to new research published on Wednesday.

A six-month unpaid internship would cost a single person living in London a minimum of £5,556, according to research by the Sutton Trust, which argues that interns who work for more than a month should be paid at least the minimum wage of £6.50 per hour.

A similar placement in Manchester would cost £4,728, or £788 a month for accommodation, bills and food, excluding transport costs which are usually paid by the employer.

An estimated 21,000 interns are working for nothing in the UK at any one time, despite widespread public concern about them giving huge advantage to young people from wealthy families better positioned to work for free. A newly published Ipsos Mori poll for the Sutton Trust says 70% of people aged 16-75 in England agree unpaid internships “are unfair because only people from wealthy families are likely to be able to work for a significant period without pay”.

Examples of unpaid internships uncovered in the research included an intern for an MP, working for six months doing administration, correspondence, diary management, fundraising and campaigning. The only payment was “reasonable” travel expenses. Another intern placement was working on the shopping and style desk for a popular magazine. Duties included writing for the section, tweeting, blogging, research and PR liason, and payment was a £2-a-day subsidy for food and a travelcard.

Internships do not have any legal status on their own, but according to the Department for Business, Innovation and Skills (BIS), which advises on employment rights and pay for interns, an intern is entitled to at least the national minimum wage if they qualify as a worker, which is someone who works for set hours – whether they want to or not – and is to all intents and purposes performing a job that a worker would be paid for.

Lee Elliot Major, director of development and policy at the Sutton Trust, said: “Unpaid internships are increasingly the gateway to a job in the most competitive professions. But as today’s research shows, the cost of taking on an internship without pay is beyond the means of the vast majority of individuals.

“Paying all interns who work for over a month the minimum wage would significantly improve access to these placements for those from more modest backgrounds, offering them a stepping stone into many coveted jobs, thus increasing social mobility.”

The Sutton Trust is campaigning for all internship positions to be advertised publicly and wants recruitment processes to be fair, transparent and merit-based.

The government’s social mobility and child poverty commission 2012 report on fair access to the professions reported that 63% of cultural and creative, 56% of media-related, and 42% of financial and professional services internships advertised on the Graduate Talent Pool website were unpaid.

As growing numbers graduate, relevant work experience is increasingly important for accessing graduate jobs. The 2012 report also noted that general work experience was regarded as vital by employers, and that access to jobs was often dependent upon specific experience with a given company. The Association of Graduate Recruiters, for example, estimated that “at least half of entry-level vacancies at City investment banks and leading law firms are likely to be filled by graduates who have already completed work experience with the employer”.

Libby Page, who did four unpaid internships before finding paid work, is a campaigner for Intern Aware which has been central to efforts to highlight exploitation. As a result of their work 100 companies were referred for investigation by HM Revenue and Customs because of concerns they were breaking the law through their use of unpaid interns. Other employers have agreed to pay former interns hundreds of pounds after they petitioned to be paid the minimum wage.

“This report gives a clear indication that although things have been changing, there’s still a lot of work to go,” she said.

“When I started campaigning there was a stronger feeling that people were really scared to talk about it. I was worried I was going to be blacklisted. People are speaking more openly now.

“But the research shows there are still a lot of people who are not being paid for their work. And it’s not just the young people doing those placements, it’s the people who are cut off from those opportunities that is the big problem.”

A BIS spokesman said the government was committed to cracking down on employers who break the law by not paying the minimum wage, and would name and shame offenders.

“Leaving education and getting a job for the first time can be daunting for any young person. Internships can provide an important first step and are often a valuable way of helping young people start work. They should be open to everyone in a fair and transparent way.

“Anyone who is a worker is entitled to be paid at least the minimum wage, including if they are an intern. Anyone who feels they are being exploited should contact the Pay and Work Rights Helpline on 0800 917 2368. Every complaint is investigated and any intern calling that number will prioritised by HMRC.”