In his column in the Daily Telegraph, the former Number 10 joint chief of staff said Mrs May had “blocked the proposal”.
He added that the vans, which appeared for a brief period in the summer of 2013, were approved while the then home secretary “was on holiday”.
Mr Timothy said Mrs May “killed off the scheme later that year, but by then the damage had been done”.
But in an answer to a written parliamentary question in 2016, it was stated that the vans “pilot” was “authorised by former immigration minister Mark Harper”.
It added that the “former home secretary, Theresa May, was informed of the intention to pilot this campaign”.
In addition, Business Insider reported it had been told by a former senior official at the Home Office that Mrs May personally intervened to ensure the language on the vans was “toughened up”.
The official also claimed that Mrs May spoke to aides about the vans while she was away.
Bloomberg, meanwhile, said it had seen Home Office emails which appeared to contradict Mr Timothy’s account.
It said that in March 2013, four months before the vans were sent out, “both May’s office and her special advisers, who at the time included Timothy, were sent plans for Operation Vaken (the name of the scheme), including images for the publicity”.
The publication added: “On March 13, May’s private secretary at the time, Matthew Bligh, replied, summarising the comments of ministers in the department.
“Far from objecting to the plan, May’s only objection at that point was that it might look too soft on illegal immigrants.”
Political commentator Dan Hodges said it was “another example of the increasingly dangerous intolerance we’re witnessing in our public discourse”.
Mr Timothy and his co chief of staff, Fiona Hill, resigned last June after Theresa May lost her majority in the General Election.