Official accounts appear to contradict claims by Theresa May’s former special adviser that she attempted to block controversial “go home” vans telling illegal immigrants to leave the country.

Nick Timothy, who was May’s adviser at the Home Office and her chief of staff at No 10 until July, said she had intended to block the rollout of advertising vans that said those in the UK illegally should “go home or face arrest”.

The vehicles, which were driven around six London boroughs with areas of high migration, became notorious as part of May’s “hostile environment” strategy aiming to crack down on illegal immigration.

Timothy claimed May had unfairly received a lot of flak for the vans. “In fact, she blocked the proposal but it was revived and approved in a communications plan while she was on holiday,” he wrote in a column for the Telegraph. “She killed off the scheme later that year but by then the damage had been done.”

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However, in a reply to the Labour MP Tulip Siddiq in 2016, the immigration minister, Robert Goodwill, said the pilot scheme, which ran between July and August 2013, had been approved by the prime minister when she was at the Home Office.

“The pilot to use the AdVans referred to was authorised by former immigration minister Mark Harper,” he said. “The former home secretary Theresa May was informed of the intention to pilot this campaign.”

May also suggested she had approved the plans when she was at a home affairs select committee in October 2013 and hinted the pilot could be extended – though it was eventually pulled.

Asked if the vans were her idea, May said: “It was part of a package that was looked at and agreed that this package would be put forward. If you are saying to me, chairman, did I say to them, ‘I think it would be a jolly good idea to have vans going round the country,’ no, it was not my initial idea. The package was brought forward and looked at, and there are variety of elements to it.”

May’s attitude was also revealed in Home Office emails from 2013, reported by Bloomberg, which show May’s office were informed in March that year about plans for the vans and suggested they were too soft on illegal immigrants.

The emails include a reply from Matthew Bligh, May’s private secretary at the time, who wrote: “The home secretary has commented that it is right to advertise enforcement action but we should not be advertising that we will pay people to leave, which is the effect of the proposed advertisements. Please can officials consider how the material can be revised to get the messaging right and not expose the agency to criticism for giving tax payers’ money to illegal migrants?”

In late March, officials then sent revised publicity images “in response to the minister’s and home secretary’s steers”, according to Bloomberg.

On Thursday, No 10 did not back Timothy’s claims that May had wanted to block the vans. “You’ve seen the prime minister’s words on various occasions, at the home affairs select committee and other places, we’ve got nothing more to add to that, it remains the position,” a spokesman said.

Liberal Democrat sources also cast doubt on the version of events surrounding the approval of the “go home” vans, suggesting that advertising budgets were very constrained by austerity measures and campaigns were signed off months in advance with ministerial approval.

One source said the Lib Dems had been given the opportunity to block the signoff, but claimed the email was sent by officials on a Friday afternoon during a staff leaving party and the message was missed.

Jeremy Browne, the Lib Dem home office minister during the period when the vans were approved, suggested after he was sacked in 2013 that his failure to block the pilot had been part of the reason behind the reshuffle, but insisted in an interview with the Times that he was not told.

“I wasn’t copied into the paper that was circulated in the Home Office. Now whether that was due to a deliberate political decision or was an administrative oversight, who knows?” he said. “It seems unduly critical to be blamed for not stopping something I didn’t know about.”

Speaking on Thursday, Nick Clegg, the former deputy prime minister, said May had “immature and overexcitable special advisers” but added that was not an excuse. “As secretary of state she bears responsibility of what goes on in her department,” he said.

The vans were scrapped after the pilot scheme, which ran during MPs’ Easter recess between 22 July and 22 August 2013. A final Home Office report revealed only 11 people had left the country as a direct result of the ads.

The evaluation report revealed that 1,561 text messages were sent to the hotline which offered to help illegal migrants return to their home countries, but 1,034 were hoaxes, which took 17 hours of staff time to deal with.

May said at the time the vans had been “too much of a blunt instrument”.