In case you were in any doubt, the prime minister and his chief aide, Dominic Cummings, more than give a damn about who runs the BBC - and intend to have a significant influence over the appointment of a new director general.

The current chairman, David Clementi, showed acute political sensitivity in persuading Tony Hall to stand down as Director General earlier than he would otherwise have done.

The point is that Clementi's own term of office ends in just over a year. And had Hall stayed in post longer than that, Johnson and Cummings would have selected a new chair to replace Clementi with the express purpose of ensuring that Hall's successor would not be hostile to their ambitions for wholesale reform of how the BBC is funded and what it does (under the rules, the government chooses the BBC chair, the BBC board chooses the DG).

So Clementi appears to have won a tactical battle over Downing Street, by taking control of the process to replace Hall. Even so Cummings and Johnson have not surrendered and intend to have a voice.

Lord Tony Hall has quit as BBC Director-General. Credit: PA

Here is what I know.

According to a well placed Downing Street source, if the BBC's board and Clementi "try to put someone like Purnell in [as DG], we will put in a chairman whose first job is to fire him...The likes of Purnell [would be] 'dead on arrival'".

This is a reference to the former Labour minister, James Purnell, who is the BBC's director of radio. It is not clear yet whether Purnell is a candidate to succeed Hall.

And what is more, Cummings and his colleagues are actively searching for possible Director General candidates they regard as sympathetic to their aims for the corporation.

All of which makes Clementi's task all the trickier - because although he is duty bound to protect the BBC from political interference and to appoint a Director General perceived as best placed to protect the corporation's independence, he is also aware that the government has the power to significantly change the scope of what it does and how it is funded through future negotiations on its charter and licence fee.

For what it's worth, Cummings' and Johnson's views on what they think is wrong with the BBC are not exactly clear, other than that they want "big change".

They've said they think non payment of the licence fee should cease to be a criminal offence. And they have raised the question whether in the era of streaming - Netflix, Apple et al - the BBC should move to a subscription model, and away from being funded through what is in effect a poll tax.

I also understand Cummings is convinced that "the BBC cannot keep on cannibalising everyone else's websites" - which the BBC would furious deny that it does.

And then there is the thorny question of what the duty to be impartial means for the BBC's important news service - with the BBC and Johnson completely unaligned on whether the corporation was impartial during the election.

All of which means Clementi needs to find the broadcasting equivalent of Henry Kissinger to replace Hall. It won't be easy.