The probe will receive "gists and quotes" of communications from former prime minister Blair to ex-president Bush in the run-up to the conflict in 2003, inquiry chief John Chilcot said in an official letter.
But Bush's replies will not be included in the report, which is examining Britain's involvement in the war, Chilcot said.
"I am pleased to record that we have now reached agreement on the principles that will underpin disclosure of... Communications between the UK Prime Minister and the President of the United States," Chilcot said in his letter to Jeremy Heywood, the British government's top civil servant.
Chilcot, himself a former civil servant, said "detailed consideration" of the agreed material had already begun.
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"Consideration will be based on the principle that our use of this material should not reflect President Bush's view," he said.
The inquiry was set up in 2009 and was expected to report in 2010. The last public hearings took place in 2011.
But disagreements over the publication of some 25 written notes from Blair to Bush and more than 130 records of conversations have been the biggest factor in delaying the findings of the inquiry.
"The Inquiry intends to submit its report to the Prime Minister as soon as possible," Chilcot added.
He did not give a date but reports earlier this year said it was expected before the end of 2014.
Blair insisted earlier this week that he was not to blame for the delay in publication of a report which could heavily criticise his handling of the war.
A total of 179 military personnel were killed in Britain's six-year involvement in Iraq.