The Prime Minister personally promised Sir James he would "fix" an issue over the tax status of his workers after he was directly lobbied by the entrepreneur, who was seeking to build ventilators at the height of the coronavirus crisis.
Mr Johnson said he was "happy to share all the details" of the exchanges as there was "nothing to conceal".
Labour leader Sir Keir Starmer suggested it was "one rule for those that have got the Prime Minister's phone number, another for everybody else".
The BBC said it has seen a series of text messages between Mr Johnson and Sir James after the businessman was unable to get the assurances he was seeking from the Treasury.
The exchanges took place in March last year at the start of the pandemic, when the Government was appealing to firms to supply ventilators amid fears the NHS could run out.
Sir James, whose firm is now based in Singapore, wrote to the Treasury asking for an assurance that his staff would not have to pay additional tax if they came to the UK to work on the project.
But when he failed to receive a reply, the BBC said he 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.
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."
When Sir James then sought a further assurance, Mr Johnson replied: "James, I am First Lord of the Treasury and you can take it that we are backing you to do what you need."
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.
Labour has claimed that there is "evidence" that Mr Johnson may have breached the ministerial code with the texts.
A spokesman for Sir Keir told reporters: "There is evidence that the code may very well have been breached and we will be following that up in the coming days."
Downing Street however insisted that Mr Johnson informed his officials of the exchanges in a "timely manner" in line with the provisions of the code.
"The Prime Minister abides by the ministerial code. He alerted officials after his contact with Dyson and then that passed on to officials to work up the advice," the Prime Minister's official spokesman said.
Challenged about the issue at Prime Minister's Questions, Mr Johnson said: "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."
SNP Westminster leader Ian Blackford called on Mr Johnson to reveal "how many more Covid contracts he personally fixed" and publish all personal exchanges on these contracts.
The Prime Minister said: "There's absolutely nothing to conceal about this and I am happy to share all the details with the House, as indeed I have shared them with my officials immediately."
Sir Keir compared the way Mr Johnson responded to Sir James' concerns with the Government's treatment of steelworkers, nurses and three million self-employed people who have been left out of coronavirus support schemes.
Workers at Liberty Steel face uncertainty following the collapse of financial backer Greensill Capital - a firm former PM David Cameron lobbied ministers on behalf of - but "the Prime Minister hasn't fixed that - in fact, he has done nothing to help steelworkers".
Nurses "didn't get a text from the Prime Minister, they got a kick in the teeth" with a below-inflation NHS pay rise.
On the self-employed, there was "an open door for those with the Prime Minister's number, a closed-door to the three million".
Sir Keir said: "What this shows once again is the extent of the sleaze and the cronyism that is at the heart of his Conservative government."
Mr Johnson insisted the Government "is doing the right thing at the right time" and "we take the tough decisions to procure thousands - tens of thousands - of ventilators in record time, which apparently he (Sir Keir) now opposes".
Sir James said he was "hugely proud" of his firm's response in "the midst of a national emergency", and that he would "do the same again if asked".
He told the BBC: "When the Prime Minister rang me to ask Dyson to urgently build ventilators, of course, I said yes.
"Our ventilator cost Dyson £20 million, freely given to the national cause, and it is absurd to suggest that the urgent correspondence was anything other than seeking compliance with rules, as 450 Dyson people - in UK and Singapore - worked around the clock, seven days a week to build potentially life-saving equipment at a time of dire need.
"Mercifully, they were not required as medical understanding of the virus evolved. Neither Weybourne (Dyson's holding company) nor Dyson received any benefit from the project; indeed commercial projects were delayed, and Dyson voluntarily covered the £20 million of development costs."
Sir James also said his company had not claimed "one penny" from governments in any jurisdiction in relation to Covid.
Reporting by PA