Greater Manchester Mayor Andy Burnham said Labour needs a leader from the "mainstream tradition" as the party has become too London-centric.
The former Cabinet member in Tony Blair's government said Jeremy Corbyn's party could revive itself after last week's crushing election defeat but there were "no guarantees".
He suggested New Labour had been "timid" and may have wasted some of its political capital during three successive election victories from 1997 to 2010 and some of the radicalism of a Corbyn agenda could be kept.
Mr Burnham said the country was in a "pretty poor state", with Brexit masking big issues, and he questioned the Conservative Government's newfound commitment to the North.
He said: "So the grounds are there for Labour to revive but it needs to revive in the mainstream tradition.
"And what I mean by that is a leader in the mould of John Smith, Clement Attlee, people who understood that Labour had to both yes be radical in the change we want to make, but also have a plan that can be acceptable to the majority of British people and we seem to have forgotten that.
"We had to learn that lesson in 1983 and I think we are going to learn it all over again in 2019.
"The Labour Party has always relied on two constituencies: Traditional voters in working-class communities and then a university-educated liberal left. Now it's absolutely possible to have policies that appeal to both.
"I remember Tony Blair famously talking about 'tough on crime, tough on the causes of crime', and that was a classic appeal to both of Labour's constituencies.
"And we've lost the art of that in recent times it seems to me, like now we are only an echo chamber for the liberal left, and that has got to change."
Mr Burnham has been closely linked to Wigan MP Lisa Nandy, who is possibly in the running for the Labour leadership.
However, he refused to name a preferred choice, with only Emily Thornberry officially announcing that she is running.
Mr Burnham said his "test" for the new leader would be someone who understands why Labour lost and the need to change the "London-centric nature of the party".
He also said he wanted a leader who advocated devolution, to take power out of Westminster to the regions, he said.
But he declined to echo the call of Mr Blair, who in a speech on Wednesday called for Labour to ditch the "sectarian hard-left" within the party.
He added: "I heard Keir Starmer this morning saying that some of the more bold, radical nature of what Jeremy Corbyn brought does need to be kept.
"Because there was a criticism of me and others who served with Tony Blair and Gordon Brown, everything had become too timid, lacklustre, managerial, and I understood that."