Also in the running is Boris Johnson's comms chief Guto Harri, who was in talks to be the Conservative Party's comms director back in 2007, before Coulson was appointed to the role.
Coulson resigned from his high-profile post this morning. In a statement, he said that continued coverage of the News of the World phone hacking scandal prevented him from carrying out his duties effectively and that he would leave ‘within the next few weeks’.
Labrokes today installed Birrell as the 2.5/1 favourite to replace Coulson. The former Independent deputy editor is highly regarded by David Cameron having been one of his key speechwriters during the 2010 general election campaign.
Downing Street insiders told PRWeek that Number 10 strategy director Steve Hilton was pushing for Birrell to get the job.
'Hilton is promoting Birrell’s case and is close to Cameron personally,' said one source.
But the source added: 'The feeling in the press office is that they need someone with either broadcast or tabloid experience. George Osborne is likely to share that view. Don’t expect a swift announcement.'
Brogan, former political editor of the Daily Mail, is at odds of 5/1 with Ladbrokes, while Harri is at 7/1 - along with former Sun political editor Trevor Kavanagh, now a columnist for the tabloid.
The Times’ Danny Finkelstein, a former Tory staffer, is at 9/1 and George Pascoe-Watson, who replaced Kavanagh as the Sun’s political editor, is at 11/1. Pascoe-Watson is now a director at Portland.
Also mooted as possible replacements for Coulson are former Sun editors David Yelland (15/1) and Rebekah Brooks (26/1). Bookies have also suggested that The News of the World's David Wooding is a contender for the job.
However, in the wake of the furore over phone hacking, and Rupert Murdoch's ongoing BSkyB takeover bid, the former Downing Street comms director Alastair Campbell warned: ‘Cameron would be unwise to hire another NewsCorp journalist.’
Amid the speculation over who would succeed Coulson, senior journalists said the former tabloid editor had performed well in the role.
The Independent’s Steve Richards said: 'It’s one of the most difficult, weirdest jobs in politics, this projecting a case to the media 24 hours a day, and he was quite good at it. David Cameron's popularity within the media noticeably improved when he got that job, so I don't blame David Cameron for putting him in there.’
The BBC’s political editor Nick Robinson said that Coulson provided a much needed common touch for the Tory leader.
He commented: ‘[Cameron] has come to rely on Andy Coulson for a connection with the real world.
‘Coulson is an Essex boy and proud of it, a man not from the old Etonian background of David Cameron and many of his closest allies.
‘It was particularly valuable to him to have someone in the room who said, "That's not how it'll play in the real world, that's not how the red-top newspapers will see it, that's not what matters to most ordinary voters".’
But The Daily Telegraph's Peter Oborne said Coulson should never have been hired at the outset.
Writing on the Telegraph website, he said: 'The key issue... is not what this episode tells us about Andy Coulson. It is what it says about David Cameron, and his decision to appoint him in the first place.
'In particular Cameron’s judgment is under scrutiny. Cameron will at some stage need to come up with a full explanation of why he decided to appoint Coulson as Downing Street director of communications. What checks did he make? What enquiries did he make of Coulson himself?'