The former News International executives' potentially explosive evidence could overshadow David Cameron's efforts to relaunch the coalition's programme after bruising local election results for the Conservatives and Lib Dems.
Mrs Brooks is likely to disclose further details about her close relationship with the Prime Minister, while Mr Coulson will speak about how he came to be appointed the Tories' top spin doctor.
Eight Cabinet Ministers - including Mr Cameron, Deputy Prime Minister Nick Clegg and Culture Secretary Jeremy Hunt - were last week given the right to see Leveson Inquiry documents and witness statements in advance.
Lord Justice Leveson agreed to make them "core participants", people who have a significant interest in the hearings or may face criticism. Mr Cameron and Mr Clegg are expected to relaunch the coalition's agenda at a joint Downing Street press conference on Tuesday ahead of the Queen's Speech on Wednesday.
But Mr Coulson's appearance before the press standards inquiry on Thursday will revive awkward questions about the Prime Minister's decision to make the former News of the World editor his communications director.
Mr Cameron said last July that "with 20:20 hindsight" he would not have hired Mr Coulson in May 2007, four months after he resigned from the Sunday tabloid over the jailing of royal reporter Clive Goodman for phone hacking.
Mr Coulson, 44, became Downing Street's communications chief in May 2010 but quit eight months later, saying controversy over the hacking scandal was making his job impossible.
His Leveson Inquiry evidence will be the first time he has spoken publicly since being arrested by Scotland Yard on suspicion of phone hacking and corruption last July. He was bailed and has not been charged.
Mrs Brooks, 43, edited the News of the World and The Sun before becoming chief executive of Mr Murdoch's UK newspapers division News International in September 2009. She and her racehorse trainer husband Charlie are key members of the influential Chipping Norton set, which also includes Mr Cameron and his wife Samantha, Top Gear presenter Jeremy Clarkson, and Mr Murdoch's daughter Elisabeth and her PR guru husband Matthew Freud.
The inquiry has already heard that Mrs Brooks regularly met Mr Cameron and other top politicians along with Rupert and James Murdoch. She hosted a Christmas dinner in Oxfordshire on December 23 2010, just two days after Business Secretary Vince Cable was stripped of his responsibility for media takeovers for saying he had "declared war" on the Murdochs' News Corporation empire.
James Murdoch has confirmed to the Leveson Inquiry that he had a "tiny side conversation" with Mr Cameron before the meal about Mr Cable's comments and News Corp's bid to take over broadcaster BSkyB.
Mrs Brooks's wedding on June 13 2009 was attended by Mr Cameron and former prime minister Gordon Brown.
Mr Cameron was forced to admit in March that he rode a retired police horse loaned to Mrs Brooks by Scotland Yard from 2008 to 2010.
There is speculation that the Leveson Inquiry could release emails and text messages sent between Mr Cameron and the former News International chief executive.
Mrs Brooks has kept all the texts she received from the Prime Minister, of which there could have been more than 12 a day, according to Daily Telegraph columnist Peter Oborne.
Emails sent by News Corp lobbyist Frederic Michel released by the inquiry last week led to calls for Mr Hunt's resignation over claims he secretly backed the Murdochs' BSkyB takeover bid.
Mrs Brooks has twice been arrested by Scotland Yard detectives investigating allegations of phone hacking, corrupt payments to public officials, and an attempt to pervert the course of justice. She was bailed and has not been charged.
The Leveson Inquiry's lawyers will not question Mrs Brooks or Mr Coulson about anything that could prejudice the continuing police investigation into phone hacking or any potential future trials.
On Wednesday the inquiry will hear evidence from Martin Clarke, editor of the Mail Online, and from officers about police corruption investigations at Scotland Yard and Devon and Cornwall Police.
Further testimony on Thursday will come from Lord Rothermere, chairman of Daily Mail and General Trust, which owns the Daily Mail and Mail on Sunday.
Mr Cameron set up the Leveson Inquiry last July in response to revelations that the now-defunct News of the World hacked murdered schoolgirl Milly Dowler's phone after she disappeared in 2002.
The first part of the inquiry, sitting at the Royal Courts of Justice in London, is looking at the culture, practices and ethics of the Press in general and is due to produce a report by October.
Lord Justice Leveson indicated last week that the second part, examining the extent of unlawful activities by journalists, may not go ahead.