Labour has accused the government of interfering with the independence of the BBC and Ofcom after the culture secretary demanded that the media regulator should scrutinise the broadcaster more closely.

Tom Watson, Labour’s shadow culture secretary, said Karen Bradley had made a “serious mistake” by writing to Ofcom to call for the media regulator to set more quotas for the BBC’s radio and TV content and to hold the broadcaster to account over the diversity of its on-screen and off-screen workforce.

Watson warned that the BBC’s political independence would be questioned “if elected politicians try to bully its regulator into changing the rules”.

The Labour MP added: “For a secretary of state to try to influence Ofcom in such a heavy-handed way is a serious mistake. I hope Karen Bradley will realise, on reflection, that she should let Ofcom get on with its job and get on with her own.”

In a letter sent at the end of last month, Bradley expressed a collection of concerns about how Ofcom plans to regulate the BBC, claiming these had been raised “by a number of stakeholders”. The culture secretary questioned the lack of quotas for the BBC’s radio content – such as the breadth of Radio 1’s playlist – as well as future requirements for arts, music and religious programming on BBC1 and 2.

She also said that the government wanted the BBC to be the “leading the way on both on and off-screen diversity in equal measure” and expects Ofcom and the BBC board to “hold [the BBC] to account for delivering in this important area”.

The minister’s letter drew a firm response from the chair of Ofcom, Dame Patricia Hodgson, who said in a reply to Bradley that it was important that the regulator was “transparently independent”.

The work of Ofcom is in particular focus because it is also advising the culture secretary on whether the takeover of satellite broadcaster Sky by Rupert Murdoch’s 21st Century Fox should be allowed.

Ofcom became the first independent regulator of the BBC earlier this yearunder a new royal charter agreed between the government and the broadcaster. The media regulator published a draft operating licence that outlined how it plans to monitor the corporation. A consultation on these plans – which closed on 17 July, a week before Bradley’s letter – has generated heavy criticism of Ofcom for not doing more to improve the diversity at the BBC.

Campaigners want Ofcom to set the BBC targets for increasing the number of BAME (black, Asian and minority ethnic) staff working behind the scenes.

Asked about the letters, a spokesperson for Bradley’s Department for Digital, Culture, Media & Sport said: “We are committed to ensuring that the reforms introduced in the new charter are in place. Following a consultation on the BBC’s operating licence, the secretary of state wrote to Ofcom to reiterate that the key reforms must be reflected.”

An Ofcom spokesperson said: “As the BBC’s independent regulator, we’re proposing clear rules to ensure the BBC is distinctive and sets itself apart from other broadcasters. We will carefully consider all responses to our public consultation, before reaching final decisions in the autumn.”