Sir James texted the Prime Minister in March 2020, with concerns over his employees' tax bills if Dyson moved to the UK to make ventilators.
The prime minister replied: “I will fix it.”
The exchanges took place at the start of the pandemic, when the Government was appealing to firms to supply ventilators amid fears the NHS could run out.
When contacted, Sir James Dyson said: "When the Prime Minister rang me to ask Dyson to urgently build ventilators, of course I said yes. We were in the midst of an national emergency and I am hugely proud of Dyson’s response - I would do the same again if asked.
Here are the texts between Boris Johnson and businessman Sir James Dyson from March 2020. Mr Johnson says he will ‘fix’ the tax system for Dyson’s employees... pic.twitter.com/cgsR0xRSeU— Victoria Derbyshire (@vicderbyshire) April 21, 2021
"Our ventilator cost Dyson £20m, 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."
He added: "Neither Weybourne nor Dyson received any benefit from the project, indeed commercial projects were delayed, and Dyson voluntarily covered the £20m of development costs.
"Not one penny has been claimed from any Government, in any jurisdiction, in relation to Covid-19."
Labour leader Sir Keir Starmer, during a visit to Hartlepool, said it was "increasingly obvious" the prime minister was involved in "sleaze".
Dyson have a site at Malmesbury
Mr Johnson, speaking to the BBC this week, responded that those who believe there is something "dodgy or rum" in his dealings with Sir James Dyson are "out of their mind".
An internal inquiry has been launched into how messages between Mr Johnson and Sir James were leaked to the press. Downing Street have so far declined to comment on "speculation" that Mr Johnson's former aide, Dominic Cummings, leaked the texts.
Two years ago Dyson faced backlash after they moved their corporate head office from Malmesbury, in Wiltshire, to Singapore.
The relocation was for commercial reasons, and operations continued from Malmesbury, because most of its manufacturing operations and customers were in Asia.