Whitehall is struggling to cope with the scale of work generated by the Brexit vote and the lack of a common strategy among cabinet ministers, according to a report about a leaked Cabinet Office memo.

The note found that departments are working on more than 500 projects related to leaving the EU and may need to hire an extra 30,000 civil servants to deal with the additional burden of work.

It identified a tendency by Theresa May to “draw in decisions and settle matters herself” as a strategy that cannot be sustained, and highlighted a split between the three Brexit ministers – Liam Fox, Boris Johnson and David Davis – and the chancellor, Philip Hammond, and his ally Greg Clark, the business secretary.

The note, leaked to the Times and said to be dated 7 November, also claimed that “no common strategy has emerged” on Brexit between departments despite extended debate among the permanent secretaries who head Whitehall departments.

In addition, it said major players in industry are expected to “point a gun to the government’s head” to get what they want after the carmaker Nissan was given assurances that it would not lose out from investing in Britain after Brexit.

A government spokesman denied the existence of an official memo, which the Times said had been drafted by an outside consultant.

The spokesman said: “This is not a government report and we don’t recognise the claims made in it. We are focused on getting on with the job of delivering Brexit and making a success of it.”

It is understood not to have been seen by ministers or commissioned as an official report by the Cabinet Office.

The note appears to be one of a number of recent leaks from the heart of government discussing dissent among senior figures about how May should approach Brexit.

The prime minister has promised to start the process of leaving the EU by the end of March next year but declined to reveal details of her approach, beyond saying there must be a bespoke deal to allow immigration controls as well as maintaining access to the single market.