The Conservative former prime minister warned that publishing only partial extracts would allow suspicions about what took place between Mr Blair and the then US president to fester.
Years of negotiations over the publication of the material, which includes 25 notes from Mr Blair to Mr Bush and more than 130 records of conversations between them, is understood to have been behind the delay in publication of the long-awaited Chilcot Inquiry report into the invasion.
Campaigners have branded the decision to limit the release to "quotes or gists" a whitewash.
Sir John urged Mr Blair and Labour to consider giving permission for full disclosure. He told the BBC Radio 4 Today programme: "I think it is a pity the papers are going to be withheld, for several reasons.
"Firstly, they will leave suspicions unresolved and those suspicions will fester and maybe worsen.
"Secondly, in many ways I think withholding them is going to be very embarrassing for Tony Blair, not least of course because he brought the Freedom of Information Act into law when he was in government."
Sir John said there were strict rules that prevented the current Government from getting involved and insisted it was down to Labour or Mr Blair to approach the Cabinet Office, which handled the negotiations, to give the go-ahead.
"Mr Blair could, the previous Labour government could, and maybe in their own interests they should think about that," he concluded.
Rose Gentle, whose 19-year-old son Gordon was killed in Iraq in June 2004, said she was "sickened" by the decision to publish only selected sections and believed Mr Blair would "walk away from it with a smile on his face".
"How will the families get to know the truth?" she asked. "We are just shoved aside."