Scotland Yard is to be asked to investigate misconduct allegations involving London mayor Boris Johnson and the Transport for London (TfL) procurement process behind the capital’s proposed garden bridge.

Campaigners are to submit a formal complaint to the Metropolitan police over allegations of “malfeasance in public office” regarding the mayor and the decision to award designer Thomas Heatherwick and the engineering firm Arup lucrative contracts for work on the controversial bridge across the Thames.

The central allegation is that the procurement process was rigged and that Heatherwick and Arup had been lined up to win the contracts before tenders were issued. It has now emerged that, just days before the invitation to tender was announced by Transport for London, Johnson – who is the chairman of TfL – met Heatherwick privately.

The mayor’s private diaries, seen by the Observer, reveal that on 1 February 2013 Johnson had a “meeting with Thomas Heatherwick”. On 13 February, TfL invited Heatherwick Studio to tender for the project along with two other firms, a process that he went on to win, despite appearing to have the least relevant design experience.

Heatherwick scored more highly than other firms in the crucial “relevant design experience” category, even though he had designed just one bridge, while competitors had designed up to 25. Additionally, despite submitting the highest quote, the bid was judged by TfL as the most “economically advantageous”.

Facebook Twitter Pinterest Mayor of London Boris Johnson. Photograph: James Gourley/Rex

A further cause of disquiet is that another key meeting, in which Johnson first raised the possibility of building a garden bridge with TfL executives, appears not to have been recorded in the mayor’s official monthly records.

It has emerged that in early 2013, Johnson met then TfL commissioner Peter Hendy and its director of strategy, Richard De Cani, to determine the feasibility of a living bridge concept. Although Johnson is meant to publish a monthly report to the London assembly listing his key decisions and activities, the meeting that effectively gave the go-ahead to a scheme that would soon lead to £30m of TfL money being allocated to the garden bridge is omitted.

Elsewhere, a senior transport source with knowledge of the process claimed the TfL board was not granted an opportunity to make a decision on the garden bridge, describing this as “highly unusual” for a project of such significance.

According to TfL papers, it appears that the first time the project was even mentioned to the board was in July 2013, when Hendy informed members: “We have appointed Thomas Heatherwick and Arup to develop plans for a new pedestrian crossing of the Thames.” Normally, even modest transport initiatives require approval by the TfL board.

Pressure on Johnson has been amplified by a former UK government adviser delivering a withering assessment of the process to appoint Heatherwick and Arup, although there is no suggestion either firm behaved improperly.

“TfL broke the law, simple as that, and it is a great shame no other bidder has challenged the decision in the courts,” said Peter Smith, formerly procurement director for the Department of Social Security and for NatWest.

Vital material relating to the evaluation process has either been lost or destroyed by TfL, meaning that the official audit report into the procurement processes of the bridge design contained no discussion about how Heatherwick scored top marks. Critics also ask why just one TfL employee evaluated the technical bids – a situation described as extremely poor practice by Smith. There are also questions over how Arup won its contract, and why it was asked to resubmit its bid while other firms were not.

The garden bridge was the idea of actress Joanna Lumley, a childhood friend of Johnson, who revealed that Heatherwick would be “happy to work on the bridge” as long ago as 2004.

The mayor’s diaries also show that he and TfL officials met Lumley in September 2012, while transport executives also met her in July and December that year.

TfL said it was “satisfied” that the procurement processes were “fair and transparent”. It added: “An extensive and thorough review was undertaken by a separate audit team, which concluded the procurement of designers for the garden bridge was acceptable in relation to the selection of bidders and there is no evidence the processes did not provide value for money.”

The mayor’s office declined to comment. Heatherwick Studio and Arup were contacted for comment, but had not done so at time of going to press.