In a letter to Cameron published today, inquiry chairman John Chilcot said the two-million-word report would be completed by mid-April 2016, ready for the authorities to conduct "national security checking".
"I consider that once national security checking has been completed it should be possible to agree with you a date for publication in June or July 2016," he said.
In his reply, Cameron welcomed the setting out of a timetable for the report's publication, but criticised its contents.
"I recognise that you have a significant task, but would welcome any further steps you can take to expedite the final stages of the inquiry."
Also Read
Cameron also said that the government aimed to complete the security checks within two weeks.
The Chilcot inquiry was set up in 2009 by prime minister Gordon Brown - the successor of Tony Blair, who led Britain into the 2003 conflict - and was originally due to report within a year.
The report is expected to highlight how Britain's involvement in Iraq - particularly questions over whether Blair's government "sexed up" a dossier on Iraq's weapons of mass destruction to make the case for war - remains the subject of heated debate.
Blair had previously denied that the delay was linked to a process in which individual witnesses such as himself are given the chance to respond to criticism of them in the report.
"For people like myself, when this inquiry finally comes out, it gives me also frankly an opportunity to come out and say 'this is why I did what I did', 'this is why I believed it was the right thing to do'," he told the BBC.
"I apologise for the fact that the intelligence we received was wrong," he told CNN in an interview.