Mr Michel has claimed he used “JH” as shorthand for Mr Hunt’s office in general, and Mr Hunt has claimed Mr Smith told News Corp far more than he should have done.
At the time the email was sent, the Metropolitan Police was six months into its ongoing investigation into phone-hacking at the News of the World, which later led to the arrest of Mrs Brooks.
In the same email to Mrs Brooks, Mr Michel states: “Hunt will be making references to phone-hacking in his statement on Rubicon [News Corp’s codename for the BSkyB bid] this week.
“He will be repeating the same narrative as the one he gave in Parliament a few weeks ago.
“This is based on his belief that the police is (sic) pursuing things thoroughly and phone-hacking has nothing to do with the media plurality issue [which the Competition Commission had been asked to examine].
“It’s extremely helpful.”
Mrs Brooks replied: “When is the Rubicon statement” and Mr Michel replied “probably Wednesday”.
Mr Hunt did, indeed, make his statement to Parliament on Wednesday, June 30, when he said the bid could go ahead subject to one final brief public consultation.
After the parliamentary statement Mr Michel texted Adam Smith to say “Think we are in a good place, no?” Mr Smith replied: “Very, yes. Jeremy happy.”
Mr Hunt faced calls for his own resignation last month after Rupert Murdoch released dozens of emails showing the extent to which Mr Smith was giving News Corporation prior warning of developments in the Government’s process of scrutinising its proposed takeover of BSkyB.
The email from Mr Michel to Mrs Brooks also discloses Mr Hunt’s thoughts on the workings of a parliamentary committee on privacy, set up earlier that month in the wake of a rash of celebrity super-injunctions.
“On the issue of the Privacy Committee,” it states, “he supports a widening of its remit to the future of the press and evidence from all newspaper groups on the regulatory regime.
“He wants to prevent a public enquiry. For this, the Committee will need to come up with a strong report in the autumn and put enough pressure on the PCC to strengthen itself and take recommendations forward.”
A spokesman for the Department of Culture, Media and Sport said: "Jeremy Hunt will respond to this when he gives his evidence to the Leveson Inquiry in due course. He is confident that his evidence will vindicate the position that he has behaved with integrity on every issue.
“It has already been made clear that when Fred Michel has claimed in emails to be speaking to Jeremy Hunt that was not the case.
“On 11 July 2011 Jeremy Hunt wrote to Ofcom for further advice about the impact of phone hacking on the BSkyB bid.”
Email from Frederic Michel to Rebekah Brooks: