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.
"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.
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The Guardian reported that Blair was "relaxed" about the publication of his correspondence with Bush.
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.
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.