The BBC revealed on Wednesday that the inventor had texted Mr Johnson in March last year to seek assurances that his staff would not face changes to their tax situation if they came to the UK to help make ventilators.
“I can confirm that, yes, we have instructed the Cabinet Office to look into this,” the Prime Minister’s official spokesman said.
“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.”
He also said Downing Street would publish correspondence between the two men “shortly”, adding: “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.”
Business Secretary Kwasi Kwarteng rowed in behind the PM today, arguing that Mr Johnson “averted an even greater crisis” on ventilators through his correspondence with Sir James.
The Secretary of State told Sky News: “I think that in the real world, in reality, people are contacting ministers, contacting MPs, all the time.
“Business people are contacting MPs all the time, constituents also contact me on my phone.
“I think that in a modern democracy it’s very good that people actually can have direct access to ministers and people who are taking responsibility.”
During Wednesday’s Prime Minister’s Questions, Mr Johnson said he made “no apology” for exchanging text messages with Sir James.
The PM hit back at Sir Keir Starmer’s questions: “If he’s referring to the request from James Dyson, I make absolutely no apology at all for shifting heaven and earth and doing everything I possibly could, as I think any prime minister would in those circumstances, to secure ventilators for the people of this country.”