The Sunday Times reported that Mr Dacre was “wooed” by the Prime Minister over drinks in Downing Street in February, and is now in talks with Number 10.
During his editorship of the Daily Mail, Mr Dacre was critical of the BBC – which he would regulate if he was made Ofcom chairman.
The Sunday Times also reported that Lord (Charles) Moore, the former editor of the Daily Telegraph, has been asked to take up the post of BBC chairman.
Ofcom chairman Lord (Terence) Burns is due to leave before the end of the year, while BBC chairman Sir David Clementi will stand down in February.
Lord Moore has previously criticised the criminalisation of people who refuse to pay the licence fee.
Tory MP Steve Baker said he would be "delighted" if Mr Dacre and Mr Moore were appointed as Ofcom boss and BBC chair.
He told Sky's Sophy Ridge on Sunday that it would be good because "they are Conservatives", adding: "They might actually start to look at the way the media functions and ensure there is some impartiality."
Is Paul Dacre an appointment you would welcome for the job of Ofcom Chairman?@SteveBakerHW says he would be “delighted” to see the former editor of Daily Mail appointed Chairman of Ofcom 'because he is a Conservative'.#Ridge https://t.co/Ags3OvGNKR pic.twitter.com/lYgScTDRrw— Sophy Ridge on Sunday (@RidgeOnSunday)September 27, 2020
Shadow culture secretary Jo Stevens questioned the Government’s priorities following the reports.
She told the show: “Well I think the whole idea of announcing appointments before a process has actually taken place is a bit strange and I think the public will be wondering where the Government’s priorities are on this.
“Why are they worrying and interfering in an open process and appointment for BBC and head of Ofcom, both two very senior independent public service posts that carry significant salaries? Why are the Government interfering in that sort of thing, when they should be concentrating on getting a grip on test and trace, keeping coronavirus rates under control and getting the economy back on track?”
Ms Stevens added: “I’m not going to comment on individuals being trailed as potential heads of the BBC or heads of Ofcom. When an appointment is made and announced we’ll take a position and we’ll take a view and comment on it then, but the fact is that the BBC and Ofcom are integral to our national interest, Ofcom regulates the BBC, they should be free from political interference, they should be impartial and the stuff that’s being trailed about these appointments is quite worrying.”
A Government spokeswoman said the processes are yet to open.
“We will launch the application process for the new chair of the BBC shortly,” they said. “It is an open recruitment process and all public appointments are subject to a robust and fair selection criteria.”