The information watchdog is investigating whether Downing Street covered up correspondence relating to Uber, amid accusations that David Cameron tried to protect the taxi app from tougher regulation proposed by the London mayor’s office.

The Information Commissioner’s Office said it was making inquiries about why Downing Street said it had no record of correspondence from a former No 10 adviser, Daniel Korski, with City Hall relating to Uber dated October 2015.

Documents released by Transport for London reveal that Korski exchanged emails with senior staff about Uber, which at the time was fighting proposals by the then mayor Boris Johnson’s office to bring in restrictions on how drivers could pick up new customers.

A spokeswoman for the watchdog said: “We have received a complaint and we are looking into these serious allegations.”

Theresa May and Sadiq Khan are also facing calls to take a close look at why their predecessors Cameron and Johnson backed away from measures that would have helped level the playing field between Uber drivers and black-cab drivers.

A string of Labour MPs and the taxi industry demanded a full inquiry into the previous Downing Street administration’s relationship with Uber, which, until this week, employed Rachel Whetstone, the wife of Cameron’s former chief strategist Steve Hilton, as its policy chief.

Cameron had a close association with Whetstone, who was godmother to his late son, and attended a Christmas party she threw in 2015 at the Mayfair restaurant Sexy Fish, which was also attended by the head of Uber’s PR and lobbying firm, Portland.

The taxi-hailing app firm also proved politically engaged during the EU referendum, when it sent out a message to its largely younger, metropolitan customers reminding them to register to vote.

In a separate development, the Licensed Taxi Drivers’ Association claimed that the Department for Business, Innovation and Skills under Sajid Javid encouraged the Competition and Markets Authority to weigh in to the debate about regulation of Uber.

In 2015, the CMA took issue with nine of the 25 proposals from TfL that could have harmed Uber’s business, warning that they would “impose regulation that excessively and unnecessarily weakens competition, to the overall detriment of users of taxi and private hire services in London”.

An email from BIS showed that an official asked the CMA whether it would be responding to the consultation, but the department denies that this or any other action amounted to lobbying of the watchdog on behalf of Uber.

The CMA also said it was untrue that Javid or BIS had intervened on Uber’s behalf to get it to respond to TfL’s consultation. A CMA spokeswoman said: “We’re responsible for making sure businesses compete to help people buying goods or services get the best possible deal. Taxi passengers ultimately want shorter waiting times, reduced fares and more choice. Our concern with TfL’s proposals were they could increase waiting times and lead to higher fares.

“We did not respond to the consultation because of an instruction by government. We chose to respond as we believed it could reduce competition and negatively impact taxi passengers. As an independent public body we routinely advise numerous government departments on a range of issues which could have an impact on how competitive industries are in the UK. This includes responding to consultations, where we have supported and recommended against numerous different proposals.”

The taxi drivers’ industry body also said the CMA had released correspondence under freedom of information laws, when BIS had denied that it held the exchanges.

BIS’s successor, the Department for Business, Energy and Industrial Strategy, said the emails relating to Uber had been outside the scope of the request because they did not relate to ministers, special advisers or senior civil servants and the CMA was asked a different question.

A BEIS spokesperson said: “There are clear procedures in place when it comes to answering freedom of information requests and all of those were followed. Searches were undertaken to identify any correspondence within the scope of the request – these were carried out within freedom of information rules and no relevant information was identified.”

Uber allows its users to book private taxis at about two-thirds of the price of traditional black cabs. Its business practices have faced scrutiny in cities around the world, with concerns raised about its pricing and tax arrangements, its effect on pollution and contribution to congestion.

Under Johnson, City Hall proposed forcing operators to wait five minutes between a booking and pickup, as well as making it illegal to show a map of cars immediately available to hire, with the mayor accusing Uber’s drivers of aggravating traffic and breaking the law in several minor ways.

But Johnson ended up backing away from the central proposals, reportedly after a barrage of requests from Cameron and George Osborne, the then chancellor, to ease off.

One of Korski’s emails from October 2015 revealed by TfL shows him discussing the development of a “sensible direction” for the proposals regarding Uber. In another, he lists a series of measures including a congestion charge, a new kind of licence and a cap on the number of non-green private hire vehicles, saying this is where he thinks the “answer lies”. But he omits to mention the proposed five-minute waiting time or ban on map of available cars.

Andy McDonald, the shadow transport secretary, said May and Khan should examine why proposals for tougher regulation of Uber had been dropped.

“We want to see a proper examination of it because it stinks,” he said. “Boris is coming out of this like a patsy. He is looking foolish and totally sat on. For a No 10 spokesman to apply pressure on City Hall is just unbelievable. It’s almost ‘where to start?’ with this one, but we will be continuing the fight for more transparency.

“Whilst we’ve got to not set our faces against disruptive technology, we can’t allow them to find ways around our legislation and taxation system. If we have to change the rules to catch up with them, then we should.”

An ally of Johnson denied that he had caved in to Uber, pointing to the measures that were adopted by City Hall, such as the imposition of English tests for drivers, which are being fought by the taxi app company in the courts.

A government source said they believed No 10 had opposed the original proposals to regulate Uber for policy reasons because they offended free market principles and were unpopular with consumers at a time when Zac Goldsmith, the Conservative candidate, was running to succeed Johnson as mayor.

However, the explanations are unlikely to satisfy critics of the relationship between No 10 and Uber. Steve McNamara, of the Licensed Taxi Drivers’ Association, which represents 24,000 traditional cabbies, said: “While the information commissioner’s inquiry into Downing Street’s apparent lies is also good news, it doesn’t alter the fact that there now needs to be a full parliamentary inquiry into this scandal. Everything that has gone on behind the scenes needs to be made public.”

An Uber spokesperson said: “More than 200,000 Londoners signed a petition in autumn 2015 against plans for things like five-minute minimum waiting times. They were also condemned by consumer groups, the media and even the Competition and Markets Authority.

“While some of these absurd measures were dropped, Transport for London is still pursuing proposals which would be bad for Londoners and drivers who use our app. For example, we are currently challenging through the courts plans for written English tests which TfL’s own estimates say would lead to 33,000 private hire drivers losing their livelihoods.”

Uber has faced a string of scandals about its internal culture in the US, triggered by a blogpost from a former employee alleging numerous sexual harassment incidents.

The company responded by setting up an urgent investigation into the claims. A few weeks later, the chief executive, Travis Kalanick, was caught on video in a fiery argument with one of his own drivers over exploitation.

Uber’s president, Jeff Jones, went on to quit, blaming the company’s corporate culture. “The beliefs and approach to leadership that have guided my career are inconsistent with what I saw and experienced at Uber,” he said at the time.