Karen Bradley, the culture secretary, has encouraged black and minority ethnic candidates to apply for new Channel 4 board posts despite rejecting the only non-white person from five applicants put forward by the regulator Ofcom.

Bradley came under fire last month after it emerged she had blocked a woman, later revealed to be the former Arts Council England deputy chief executive, Althea Efunshile, from the board of Channel 4, which has no BAME members and only three women.

But during questioning by the Lords communications committee on Tuesday, Bradley said there would be a transparent process for appointing the next three board members.

“There will be future appointments that will be properly advertised in a transparent way, and I would encourage applications from all candidates,” she told peers. “I want to see a diverse range of candidates appointed to public appointments. I am absolutely determined that we will see full representation.”

However, Lady McIntosh of Hudnall said: “A certain amount of damage that had been done … because of the particular way these appointments had been announced.”

Asked by Lady Benjamin what she would do to reassure female and BAME candidates that they would be considered on their own merits, Bradley said she wanted to assure any candidate they would be taken seriously. ”I will make sure the very best person gets the job,” she added.

On the future ownership of Channel 4, she said the government wanted to go through “all the options and all the scenarios”. The government is considering privatising the broadcaster.

Bradley is also overseeing the appointments to the new unitary BBC board, which is in the process of interviewing for a chairperson. She said the government had scrapped a requirement for the board to represent licence fee payers because the BBC should “prioritise the public good”.

“There was concern for having a specific remit in the board to represent the licence fee payer [because] the licence fee payer may not want what is in the public good,” Bradley told peers.

“There may be demand for high ratings programming, but ... perhaps that programming is not so popular but is actually what we want the BBC to deliver as part of that distinctiveness and the plurality of public service broadcasting.”

Earlier in the hearing, Bradley stressed she was “open-minded” on press regulation and whether to introduce laws forcing newspapers to pay both sides’ costs in libel cases if they did not join a state-backed regulator.

She said she was waiting for the result of a 10-week consultation before making any decisions on the matter or on whether the second part of the Leveson inquiry – to examine the relationship between the press and police – should go ahead.