The First Minister insists that, on what he regards as the "greatest foreign policy blunder of the modern age", only full disclosure will do and give the public confidence in the findings of the Chilcot Inquiry, which has examined at length the causes of the Iraq War.
After a four-year delay, it is now expected the inquiry will publish its report before the end of the year.
But it appears this has only been made possible because of a deal struck with the Cabinet Office not to publish the Blair-Bush exchanges in full, but just to give the "gist" of them.
Direct quotations will be kept to "a minimum necessary for the inquiry to articulate its conclusions" and agreement has also been reached that any use of material from the letters "should not reflect President Bush's views".
Sir John Major, the former Conservative Prime Minister whom Mr Blair defeated to win power in 2007, has expressed disappointment.
He has called on his former political rival to ask the inquiry to publish his correspondence with the former president because releasing only the "gist and quotes" would allow suspicions to fester, with some suspecting Mr Blair promised Mr Bush the UK would join in the military invasion come what may.
In a letter to Sir Jeremy Heywood, the Cabinet Secretary, the First Minister echoes the sentiment. He acknowledges what he calls the long-standing convention that many standard communications between and within governments are often outwith the rules of public disclosure.
But, he says: "On questions of war and peace such as this it is vital that there is such full disclosure, given that it goes to the heart of what the Chilcot Inquiry and the other various inquiries into the Iraq War were set up to establish.
"That is needed to establish whether guarantees were given by Mr Blair to Mr Bush over the UK's involvement in Iraq; whether those guarantees were verbal, written, formal or informal."
The FM insists that this "goes to the heart of the inquiry" and was one of the key charges laid against Mr Blair when the war was debated in the UK Parliament.
Mr Salmond asks: "How can people be confident that this aspect has been properly investigated if these exchanges between Mr Blair and Mr Bush are not to be released into the public domain when the inquiry reports?
"It would be quite wrong for the full correspondence not to be released given that, in my opinion, the invasion of Iraq in 2003 was the greatest foreign policy blunder of the modern age.
"It was a policy decision which resulted in the loss of countless thousands of lives and which continues to cause immense pain and suffering for people in Iraq and here in the UK, including among the loved ones of UK military personnel who were lost.
"More than a decade on from the decision to invade, it is vital the public know the answers to one central question: did Mr Blair promise Mr Bush that he would take the UK to war in Iraq come what may?"
In 2004, the SNP leader led attempts to impeach Mr Blair and put the ex-premier on trial for "high crimes and misdemeanours" in taking Britain to war in Iraq.
The campaign, backed by celebrities including the late playwright Harold Pinter, was eventually abandoned. In 2009, it was revealed the attempt to impeach the ex-PM had cost the taxpayer £14,000 in legal bills.