The pair have clashed over a number of issues – most notably the Iraq war – and Mr Corbyn's followers view his rise to Labour leader as a chance to consign the Blair years to the dustbin of history.
But while there is no love lost between the pair, Mr Blair is very much in the Labour camp at the general election, with a Boris Johnson-led Tory government at the very top of his current list of concerns.
"I'm not hiding my differences with Jeremy Corbyn," he tells me, shortly after addressing Labour members at the Blakenhall Community and Healthy Living Centre in Wolverhampton.
"But I have a deep anxiety over the damage a Conservative government will do to public services and Britain's economy."
He was in the city to support his old friend Pat McFadden, who is attempting to get re-elected in Wolverhampton South East for the fifth time.
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Mr McFadden, who served as a minister under Gordon Brown, is from the school of Labour moderates that Mr Blair describes as "decent" and "mainstream" – a side of the party that many traditional Labour politicians feel is being ostracised under the current leadership.
It is perhaps not surprising that Mr Blair, who won three elections in becoming Labour’s longest serving PM, has stopped short of endorsing Mr Corbyn.
He's unconvinced by elements of Labour's radical manifesto, questioning the practicality of some of the lavish spending plans and expressing concern over its impact on the business community.
And besides, he doesn't think Labour can win a majority.
"There are some perfectly good ambitions in there," he said about Labour's manifesto.
"About investing in public services, dealing with poverty and inequality, and we should do all that.
"But you have got to have practical policies. My view is that the risk is not that the Labour manifesto is going to get implemented, because I don't think that is what we are going to be faced with on December 13.
"A majority Labour government is unlikely if you look at the polls. People understand that risk, but it is one per cent. The risk on the Tory side... I don't think people understand at all."
That "risk" is tied to Brexit, according to Mr Blair, who when he raises his head above the parapet these days can usually be found fighting the corner for Remain.
"The trouble with Boris Johnson is that he is not reliable," he says.
"You ask the Ulster Unionists how reliable he is. He got the leadership condemning Theresa May for a deal he then signed up to.
"Now he is saying 'I've got my deal, if you vote Tory on December 12, that's it', we exit on January 31 and it is all over. But it is far from over.
"Brexit will drag on for far longer than he is letting on."
'No deal will mean serious problems'
He says he fears a big majority for Mr Johnson could still lead to a no-deal Brexit, saying: "It's a bigger risk than people think.
"If we end up with a no deal we will have serious problems with our economy."
He agrees with Labour's push for a second referendum, but admitted it was "worrying" that Labour voters were switching to the Tories over Brexit.
"You have to try and explain to people who are voting for Boris Johnson to get Brexit done, that he won't get it done," he said. "It is not going to be over."
Mr Blair says Labour should never have agreed to an election in the first place, claiming: "You should not mix Brexit up with the question of who runs the country.
"It was a mistake for the country, and it poses difficulties for people over who to vote for.
"There are people who are anti-Brexit but worry about Jeremy Corbyn, and people who are pro-Brexit but worry about the Conservatives on the NHS.
"It was never a good idea."
He describes the current state of British politics as "very difficult", but claimed the job had become far more complex as "politics has moved to the extremes".
And in an election campaign where Brexit was by far the most talked about issue, he lamented that "two of the most important things" happening in the world – the technological revolution, and the rise of China – were being completely ignored.
But despite his concerns, he is not tempted to return to frontline politics, although he says he has felt compelled to become more vocal in recent times.
"When you stop being Prime Minister you start to see the world somewhat differently, and you realise that things are changing so fast," he says.
"A country like Britain... it's a great country with great people, and you can't just put your head in the sand."