But reports said Downing Street sources are pointing the finger at Mr Cummings, who quit as Mr Johnson’s senior adviser last year following a behind the scenes power struggle in Number 10.
The Times, Daily Telegraph and Sun all reported comments from an insider naming Mr Cummings.
“Dominic is engaged in systematic leaking,” a source told The Times. “We are disappointed about that.
“We are concerned about messages from private WhatsApp groups which have very limited circulation.”
The source suggested the Prime Minister was “saddened” and Mr Cummings was “bitter” after his exit from No 10.
The leak of the texts to Sir James, in which Mr Johnson promised the entrepreneur he would “fix” a tax issue for Dyson staff working to develop ventilators at the height of the coronavirus crisis last year, was not the first time the Prime Minister’s messages have been made public.
Mr Johnson was sent a text message by Saudi Crown Prince Mohammed bin Salman as a bid to buy Newcastle United ran into difficulties last June
A No 10 source told The Sun that Mr Johnson “fears Dom was responsible for the text message leaks about James Dyson and Mohammed bin Salman”.
The Telegraph said it is understood Mr Cummings would have had legitimate access to the messages while he worked at No 10.
“If you join the dots it looks like it’s coming from Dom,” a source told the newspaper.
Mr Cummings has not responded to the accusations.
The former Vote Leave mastermind worked closely with Mr Johnson on the Brexit campaign and was a major figure in No 10 after the Prime Minister took office.
The Prime Minister stood by him after Mr Cummings found himself in the eye of a media storm after driving his family to County Durham during the coronavirus lockdown.
But Mr Cummings was subsequently ousted from No 10 amid the fallout from an internal power struggle with the Prime Minister’s fiancee, Carrie Symonds.
No 10 had initially said there would not be a probe into how the exchange with Sir James was made public, but a change of course was announced on Thursday as it said an internal inquiry will be led by the Cabinet Office.
The Prime Minister’s official spokesman told a Westminster briefing: “I can confirm that, yes, we have instructed the Cabinet Office to look into this.
“The position has changed from yesterday – it was correct at the time yesterday but, as usual, we keep things under review and we have now decided to undertake this internal inquiry.
“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.”
The spokesman confirmed the inquiry will examine the source of leaks of Mr Johnson’s private communication “as related to this issue of Dyson”.
The BBC reported that the messages between Mr Johnson and Sir James were exchanged in March last year after the businessman was unable to get the assurances he was seeking from the Treasury.
Sir James, who has changed his main address in business filings to the UK from Singapore, wrote to the Treasury requesting that his staff would not have to pay additional tax if they came to the UK to work on the ventilator project.
But when he failed to receive a reply, Sir James reportedly took up the matter directly with the Prime Minister.
He said in a text that the firm was ready but that “sadly” it seemed no-one wanted them to proceed, to which Mr Johnson replied: “I will fix it tomo! We need you. It looks fantastic.”
The Prime Minister then texted him again saying: “(Chancellor) Rishi (Sunak) says it is fixed!! We need you here.”
Two weeks later, Mr Sunak told the Commons Treasury Committee that the tax status of people who came to the UK to provide specific help during the pandemic would not be affected.
Downing Street has said it will publish correspondence between Mr Johnson and Sir James “shortly”, after the Prime Minister told the Commons he was “happy to share all the details” of the exchanges.
The Prime Minister’s official spokesman said: “The Prime Minister said in the House he’s happy to share all the details with the House, as he shared them with his officials.
“That’s what we’re working on, we’re pulling together that information, it will be published shortly.”
Meanwhile, the spokesman did not deny reports that Cabinet Secretary Simon Case advised Mr Johnson to change his phone number over concerns about the ease with which lobbyists and business leaders were able to contact him.
The spokesman told Westminster reporters: “We don’t get into details of the advice provided between a Cabinet Secretary and a Prime Minister, and so I’m not going to do that in this instance.”
