Andy Burnham has promised not to "re-run" Brexit arguments, as he said he was not proposing the UK considers rejoining the European Union.
The Greater Manchester mayor, who is expected to challenge Sir Keir Starmer for the Labour leadership if he wins the Makerfield by-election, said Brexit had been "damaging" but "Britain will be stuck in a permanent rut if we're just constantly arguing".
Burnham said last September that he would like to see the UK back in the EU within his lifetime.
Meanwhile, Sir Keir has insisted he will not set out a timetable to stand down as prime minister if Burnham returns to Westminster.
Brexit looks set to become an issue in any potential leadership contest, after former Health Secretary Wes Streeting said in a speech on Saturday that leaving the EU had been a "catastrophic mistake" and the UK should "one day" rejoin the bloc.
Streeting, who resigned from the government last week and has confirmed he would enter any future leadership race, told a conference that the "biggest economic opportunity we have is on our doorstep" as he called for a "new special relationship" with the EU.
In order to stand to replace Sir Keir, Burnham would need to win a by-election in the Makerfield constituency, which voted strongly for Brexit in 2016.
He has already been cleared to seek selection as Labour's candidate, after the current MP Josh Simons stood down to make way for him, although he still needs to be chosen by the local party.
In a speech at a conference in Leeds, Burnham said: "My view is that Brexit has been damaging, but I also believe the last thing we should do right now is re-run those arguments."
He added: "I am not proposing that the UK considers rejoining the EU.
"I respect the decision that was made at the referendum and it is going to undermine everything I have said about strengthening democracy if we don't respect that vote."
The Liberal Democrats accused Burnham of "U-turning before he's even been elected" over his Brexit stance.
Sir Keir has already been seeking closer ties with the bloc, but has stuck to Labour's election manifesto pledges to "stay outside the EU", with "no return to the single market, the customs union, or freedom of movement".
Setting out his broader pitch, Burnham told the conference that a vote for him was a vote to "change Labour".
He called for a transfer of money and resources from "a bloated national state" to "a malnourished local one", saying the country needed "serious rewiring".
"We have hollowed out councils and have created an unaccountable state, where too much is delivered by outsourced agencies outside local councillors' control," he said.
Burnham said his party's offer to voters had "not been good enough" and that "a vote for me would be a vote to change Labour".
"Labour needs to change if we are to regain people's trust," he added.
Asked if he would set a timetable for his departure if Burnham won the Makerfield by-election, Sir Keir told broadcasters: "I'm not going to do that."
He added that the by-election was "a fight between Labour and Reform" and "I will be backing 100% whoever the [Labour] candidate is".
Sir Keir repeated his insistence that he would "not walk away" from the job of prime minister.
He said he wanted to fight the next general election but recognised that "we've got to turn things" after the party's dire election results on 7 May.
It comes after a week of turbulence in his party, with five ministers resigning over Sir Keir's leadership following the results.
The Makerfield constituency was was won by Labour in 2024 with a majority of 5,399 votes over Reform UK.
There are no precise figures for how the parties performed in Makerfield during the local elections because the ward boundaries do not fully align with the constituency boundaries - but Reform won roughly 50% of the vote share in the area.
Party leader Nigel Farage has said Reform "will throw absolutely everything at" the by-election contest.
