Downing Street has announced an internal inquiry into the leak of private text messages between Boris Johnson and billionaire James Dyson over the tax status of his employees.

The prime minister promised the businessman he would “fix” the issue after personal lobbying from Dyson via texts.

Dyson was seeking assurances before he agreed to build ventilators at the height of the coronavirus crisis. In the end, Dyson never supplied any ventilators to the NHS.

Labour has demanded an urgent investigation be conducted by parliament’s liaison committee, which is made of up senior MPs.

Speaking during PMQs on Wednesday, Labour leader Keir Starmer suggested it was “one rule for those that have got the prime minister’s phone number, another for everybody else”.

The text messages were obtained by the BBC. Asked if there would be a leak inquiry, the prime minister’s spokesperson said on Thursday: “I can confirm that, yes, we have instructed the Cabinet Office to look into this.”

Downing Street also said it will publish correspondence between Johnson and Dyson “shortly”.

“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,” the spokesperson said.

“That’s what we’re working on, we’re pulling together that information.”

No.10 also did not deny reports, including in The Times, that cabinet secretary Simon Case advised Johnson to change his phone number because of concerns over the ease with which lobbyists and others from the business world were able to contact him.

The spokesperson said: “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.”