The inquiry is likely to examine the circumstances leading up to Britain's decision under Tony Blair, Brown's predecessor, to join the invasion and its aftermath.
David Miliband, the UK's foreign secretary, said in March that a probe would be held soon after almost all of Britain's 4,100 troops withdrew from Iraq by the agreed date of July 31.
Miliband has indicated that the inquiry would be held in private, but several politicians and some relatives of UK soldiers killed in Iraq have called for it to take place in public.
In an interview with Britain's Observer newspaper, Nick Clegg, the leader of the Liberal Democrats, the country's second-largest opposition party, said: "If he [Brown] holds it all, or partly, in secret and kicks the eventual report into the long grass, it will be a betrayal of all those families who lost children serving in Iraq.
"They need answers, not another Whitehall stitch-up."
Rose Gentle, an anti-war campaigner whose soldier son Gordon died while serving in Iraq, said: "We already feel that we've been lied to by the government. We don't want any more lies."
The announcement is set to come as Brown strives to reassert his badly damaged authority over his own party.
A scandal over politicians' expenses and historically bad results in European and local elections have prompted a wave of ministerial resignations and calls for him to resign.
A total of 179 British military personnel have died since March 2003 while serving in Iraq.