David Cameron will face questions on his links with former News International executives
The Mail on Sunday says Cameron will give evidence in mid-June to the inquiry, which is turning its attention to the relationship between the press and politicians. His appearance will reportedly come after Brown and Blair.
Writing on the Spectator's Coffee House blog, journalist James Forsyth says: "One ally of [Cameron] predicts that it will be an 'excruciating experience' for him. The inquiry is likely to rake over the coals of several incidents that have caused him political damage."
Other ministers said to have been asked to submit evidence before the end of this month are deputy prime minister Nick Clegg, culture secretary Jeremy Hunt, home secretary Theresa May, justice secretary Ken Clarke, business secretary Vince Cable and education secretary Michael Gove.
Gove, a former Times journalist, told parliamentary reporters at a lunch earlier this year that the wide-ranging inquiry chaired by Lord Justice Leveson could produce "a cure that is worse than the original disease" and that there is "a chilling atmosphere towards freedom of expression which emanates from the debate around Leveson".
Cameron is expected to be asked about the Conservative party decision to hire former News of the World editor Andy Coulson as head of communiations, and the prime minister's links with former News International chief executive Rebekah Brooks.
Brooks has been granted core participant status for the third module of the Leveson inquiry, as have MP Tom Watson and former MP Evan Harris, who will join politicians Chris Bryant, Simon Hughes, Tessa Jowell, Denis MacShane, Lord Prescott and Claire Ward, who have continued status.
A spokesman for the Leveson inquiry said that witnesses are not confirmed until a week prior to their appearance.