A government source told the Independent newspaper that "good progress" had been made in talks between the inquiry body and the government on declassifying many of the records of their conversations.
There had been fears that Blair -- who as prime minister committed British forces to the US-led invasion of Iraq in 2003 after forming a close bond with then-president Bush -- and the US administration would block the release of the confidential papers.
But the Independent and the Guardian both reported that a compromise had been reached that should allow the documents to be published in the next few months, with as little redaction as possible.
"There are likely to be some redactions -- but only where absolutely necessary," the government source told the Independent.
The release of the documents would allow the Iraq Inquiry to complete its final task of contacting people who are set to face criticism in the report and allow them to defend themselves.
The issue of the records of discussions between Blair and Bush has been one of the major obstacles to the progress of the inquiry, which was set up in 2009.
A letter to the government from the inquiry's chairman, former senior civil servant John Chilcott, in November said he had asked for the disclosure of 25 written notes from Blair to Bush and more than 130 records of conversations between Bush and either Blair or his successor Gordon Brown.
A spokeswoman for the Cabinet Office, which is managing the inquiry, said: "As the exchange of letters between Government and the Inquiry shows, these issues are being worked through in good faith and with a view to reaching a position as rapidly as possible.
"The inquiry should be allowed to publish its findings and we should not pre-empt the content of the report."
Britain was the second largest contributor of troops to the invasion after the United States.
A total of 179 military personnel were killed in Britain's six-year involvement in Iraq.
Former top British civil servant Lord Butler said earlier that Blair had “deliberately” excluded members of the government from the decision-making process in the run-up to the US-led invasion of Iraq.
Butler said Blair intentionally kept away the majority of the Cabinet from documents drawn up to examine the case for the Iraq war in 2003, the Civil Service World magazine reported earlier.
Blair is globally discredited for his war crimes, as under his premiership between 1997 and 2007 hundreds of thousands were killed and injured in illegal military interventions.
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