The Cabinet Office is to launch an internal investigation into the leak of Boris Johnson’s text messages with the billionaire businessman James Dyson – with reports saying that some in No 10 are accusing the prime minister’s former adviser Dominic Cummings of being the source.

The prime minister’s spokesperson said the decision had been made to launch a formal investigation into the leak, which showed that Johnson promised to change tax rules by saying: “I will fix it tomo!” It comes amid growing concern over Johnson’s use of a personal mobile phone in government.

But the inquiry will not examine a string of other leaks, including a text to the prime minister from the Saudi crown prince, Mohammed bin Salman, or leaked emails about donations solicited to cover the renovations of the prime minister’s flat.

Cummings, Johnson’s former all-powerful adviser who was booted out of Downing Street last November, was reportedly accused by No 10 sources quoted in multiple newspapers as being responsible for the leaks. In what appeared to be a coordinated attack on Cummings, the Telegraph, Times and Sun reported the same criticisms from an unnamed insider accusing him of being “bitter” about leaving government.

They said Johnson was “disappointed” and “saddened” by what they claimed Cummings was doing, complaining it was “undermining” the government. The Telegraph added Cummings would have had legitimate access to the messages while he worked at No 10, quoting an insider who said: “If you join the dots it looks like it’s coming from Dom.”

Downing Street declined to formally comment on the reports. The briefings may risk drawing attention away from the contents of the leaks, which have resulted in multiple inquiries being set up, and instead on to the blame game as to who is responsible for them.

Johnson, who has had the same phone number for a decade, is regularly texted by business leaders and politicians, sources have said. The prime minister is understood to have liberally distributed his personal number over the years.

Labour argued a leak inquiry missed the point. The opposition is expected to step up the pressure on the government, and demand that all ministers disclose their text message correspondence relating to government contracts in the coming days.

Rachel Reeves, the shadow Cabinet Office minister, called on ministers to be “vastly more transparent”. She added: “We need all ministers to offer urgent reassurances to the British public that they are not pandering to Tory friends and donors at the expense of taxpayers.”

But some ministers and special advisers fear any new rules obliging them to report every unsolicited text could “open the floodgates” and say “government by WhatsApp” is the norm across Westminster.

Others who have worked in Whitehall warned that Johnson has a “looser style” of government and may not always run all his communication through his private office. No 10 has said the prime minister always abides by the rules.

“There is a loophole that none of this stuff has to be publicly declared, as long as you are telling your officials of any government business, but if these inquiries order that everything needs to be declared, it will be mountains,” one former official said.

Another former civil servant, who was responsible for business liaison with Downing Street and other departments, said they believed Johnson’s mobile phone use demonstrated that “he just doesn’t understand that there needs to be a level of transparency and accountability” in regards to his actions.

They argued that the danger with engaging in direct correspondence was that it implied the prime minister believed “his own judgment on something is sufficient and no further information or assessment is required”.

A more proper approach, they argued, was to refer any direct lobbying to his private office and where necessary to arrange a meeting or phone call.

The row was sparked after the prime minister was revealed to have texted the pro-Brexit billionaire Dyson last year, promising that his staff would not have to pay extra tax if they came to the UK to make ventilators during the pandemic.

On Wednesday, Downing Street had said there would be no leak inquiry, only to change its mind 24 hours later.

“We have now decided to undertake this internal inquiry,” the spokesperson said. “As you would expect, we continually look at this and the position we decided today is that we want to make sure we have this internal inquiry into that.”

It will be an internal Cabinet Office inquiry and will not involve the police or the security services. It is understood the texts from Dyson were forwarded to a number of officials and aides in Whitehall.

Johnson’s spokesperson would not deny on the record that the cabinet secretary, Simon Case, advised Johnson to change his number because it was so widely known.

The prime minister promised to publish his text correspondence with Dyson, in a reply to the SNP’s Westminster leader, Ian Blackford, at prime minister’s questions, though No 10 aides have not given any timeframe for that to happen.

Labour also wrote to the chair of the liaison committee, Sir Bernard Jenkin, asking for the committee to question the prime minister on the use of his personal phone. The committee, which is made up of all the senior MPs who chair other select committees, will take evidence from Johnson before the summer recess.

There is no requirement in the ministerial code to declare lobbying approaches made by text, and no clear guidance on whether the code covers virtual meetings. However, texts to the prime minister concerning government business are covered under freedom of information legislation.

Meanwhile, a former minister has been rapped by the appointments regulator for failing to declare five external positions after leaving government.

George Freeman, the former transport minister, told the advisory committee on business appointments that it had been a genuine oversight that he failed to declare the work, including with PPE company Aerosol Shield, which intended to sell its products to the NHS.