In a new book published in the UK today, Campbell recounts a trip Blair made Down Under in 1995, when he was Britain's Labour opposition leader, and Keating was Australia's Labor PM.
During the visit, Keating - the master of the political put down - offered a range of advice to Blair, from how to handle elections, to dealing with media mogul Rupert Murdoch.
In th book, entitled The Alastair Campbell Diaries: Prelude to Power, Campbell says he was so impressed by Keating, he believed Blair could learn some lessons from him.
He says he actually asked the Aussie Premier - known for his acid-tongued attacks on opponents - to coach Blair on how to hate the Tories.
"[But] he said TB was probably right, because every time he went over the top, he went down in the polls," Campbell wrote.
John Howard 'a nobody'
In another conversation, Keating demonstrated his notorious vitriol for the-then Liberal Party leader John Howard.
Keating told Blair he believed Australian voters "would be making a terrible mistake" if they elected Howard as prime minister at the 1996 general election.
Campbell recalled Keating telling Blair that Howard was "a nobody, stands for nothing, nothing to say about the future".
Keating then advised Blair to "keep coming up with new ideas" before confessing the Australian Labor Party should never have won the 1993 election.
"But we won because we stayed ahead on policy detail and people felt we had an agenda for the future," he was said to have told Blair.
Keating also had some tips for Blair on economic issues, telling him that if he ever became prime minister he should avoid income tax hikes at all costs.
'Avoid TV debates' advice
"I've got some good advice for you about income tax - don't put it up. Ever," Campbell recounted Keating as telling Blair.
"Tony, promise me you won't raise income tax. It's death. Labour parties around the world have enough to contend with without hanging that round their necks. It's not worth it."
Just two months before Blair won the 1997 British election, Keating was on the phone again with more advice.
This time, he told Blair not to be tempted to take part in TV debates with his political rivals.
"He said, whatever you do, don't get drawn into a TV debate, because they're a pain in the ass and the underdog usually wins," Campbell wrote.
"And don't go overboard until the last two weeks. Keep a lot of powder dry till then. And don't give them the economy."
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