Andy Burnham has accused former Labour Prime Minister Sir Tony Blair of failing to understand "what's going on" in people's lives and underestimating the impact of inequality.
Sir Tony used a 5,600 word essay to argue the Labour government had "no coherent plan" for the country and had introduced policies that had held back business.
He urged Labour not to move to the left and to embrace the "radical centre" instead.
But Burnham, who is widely expected to challenge Sir Keir Starmer for the Labour leadership if he wins a by-election next month, told the Observer Sir Tony "doesn't mention inequality once" in his critique of where the Labour government has gone wrong.
"If you don't get how that's driving politics now, if you are not rooting your analysis in the fact that people are unable to live and that things that were taken for granted are no longer affordable, then you are not understanding what's going on," said the mayor of Greater Manchester.
Burnham is seeking to become an MP again on 18 June in the Makerfield by-election, on the outskirts of Wigan, in what is expected to be a closely-fought contest with Reform UK's Robert Kenyon.
Former Health Secretary Wes Streeting has also said he will stand if there is a leadership contest - but Sir Keir has said he will not walk away from the top job.
In an interview with BBC Radio 4's Today programme, Sir Tony criticised Burnham for claiming Britain has been "on the wrong path for 40 years" - a period that includes Sir Tony's 10 years in power, from 1997 to 2007.
"I hope Andy wins Makerfield, I think he's a great guy, I want to see him in Parliament," said Sir Tony.
"But you know, when he does this thing about 40 years of wasted… I mean, OK, and what, nothing good happened in that period of Thatcher with the business community, or New Labour?
"I don't think he really means that."
Burnham, who was a junior minister under Sir Tony before being promoted to the cabinet by Gordon Brown, hit back in his Observer interview, saying: "The last 40 years has given us wide inequality - that's what's responsible for the abandonment of the centre.
"People don't think the centre has delivered for them in terms of their lives, therefore they've gone further to the extremes."
Sir Tony claims in his essay that Labour has suffered from a "perennial delusion – that when we lose seats to the right the country is really signalling it wants Labour to move left".
He calls for the party to champion the "radical centre".
Asked if he considered himself to be left wing, Burnham told the Observer: "If you want to call it left wing that's fine by me.
"It's knowing where you need to take a more left solution and where you want to be pro-business.
"Blairism sometimes saw the market as always the answer. That's its problem."
In his essay, Sir Tony said he agreed with some of the government's policies, including investment in infrastructure, reform of the planning system and reducing trade friction with Europe, but said other commitments were "unwise to proceed with" given the current economic circumstances.
He pointed to new workers' rights laws, which have faced criticism from some business groups who argue they will discourage companies from hiring and hit economic growth.
Sir Tony also criticised the decision to increase National Insurance for employers, which he said had undermined business confidence.
Setting out his own vision for change, Sir Tony said Labour must remove obstacles to business growth, take action to tackle illegal immigration and harness artificial intelligence (AI).
In a detailed rebuttal of Sir Tony's essay, Torsten Bell, the former head of the Resolution Foundation think tank, who is now the pensions minister, said it was an "impressive attempt to engage with some of the big forces shaping our future".
But, he argued, "it doesn't have a project that remotely fits the time and place we are living in".
"Saying 'AI' is not the same as having a plan for Britain," he added.
In his BBC interview, Sir Tony was asked about the sources of funding for his think tank, the Tony Blair Institute, and the influence of donors, including Larry Ellison, the founder of software company Oracle.
Sir Tony said he was happy to work with Ellison because "we share the same view about this technology revolution".
He said working and accepting money from such people was "honestly to do with looking at the world and asking what the right answer is".
You can see a full list of the candidates standing in the Makerfield by-election here.
