Mr Johnson promised the entrepreneur he would “fix” the issue after personal lobbying from Sir James as he sought to build ventilators at the height of the coronavirus crisis, in a series of text messages seen by the BBC.
No 10 had initially said there would not be a probe into how the exchange 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”.
During Prime Minister's Questions on Wednesday, a defiant Mr Johnson told MPs: "I make absolutely no apology at all for shifting heaven and earth for doing everything I could... to secure ventilators for this country and save lives."
The prime minister said he was forced to act in March last year because Covid-19 was "capable of killing people in ways that we didn't understand, the only way to help them was to incubate them and put them on ventilation".
Labour leader Sir Keir Starmer took issue with the way Mr Johnson was "lobbied by a wealthy businessman and a close friend for a change in the tax rules" and questioned whether those left out of coronavirus virus support could have received the same kind of help.
"This shows once again that favours, privileged access, tax breaks for mates, they're the main currency of this Conservative government."
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