Andy Burnham drew the battle lines for the future of the Labour party on Monday as the Greater Manchester mayor promised he would “change Labour” and win back the voters the party had lost.

Burnham, who is expected to be Labour’s candidate in the Makerfield byelection, claimed it would be no ordinary campaign and said he would make it about national issues where Labour was failing, in a direct challenge to the prime minister.

But a defensive Keir Starmer said on Monday he had no intention of stepping aside should Burnham win the byelection and that he wanted to fight the next election, setting up a potentially ugly battle for the leadership.

“If I get to stand, a vote for me will be a vote to change Labour, because Labour needs to change if we are to regain people’s trust,” Burnham said.

He said his campaign in Makerfield would “show how we lift up its people and places over the next decade. It will involve action to make the basics of life more affordable, like rents, bills and fares.”

Allies of the mayor said he would use this week to set out his economic agenda and to forcefully close down issues that might derail his return to Westminster, including ruling out any imminent return to the EU and recommitting to the fiscal rules on borrowing and debt.

In comments designed to end a damaging row over Labour’s position on Europe – but which might irk some of his backers in the party – Burnham said he did not want to restart debate about EU membership.

“My view is that Brexit has been damaging, but I also believe the last thing we should do right now is rerun those arguments,” he said, a direct challenge to his would-be leadership rival Wes Streeting, who said over the weekend that the UK should seek to rejoin.

He said Britain would be stuck in “a permanent rut if we’re just constantly arguing” and said he would have a “relentless domestic focus” during his byelection run.

Burnham’s spokesperson also ruled out any further changes to Rachel Reeves’ fiscal rules, including exemptions for defence spending that he had previously considered, in an effort to reassure jittery financial markets.

In his speech to the Great North Investment Summit, Burnham said he would use the byelection as a platform to link local issues to his diagnosis of what had gone wrong for Labour and the country.

“I know what my party has offered in the past has simply not been good enough,” he said. “The loss of faith of voters across the north, so many of whom once saw us as their natural party, is our fault and nobody else’s.”

Josh Simons, the MP who resigned in order to allow Burnham to stand in his seat, officially departed the Commons on Monday, with a date for the poll expected to be confirmed on Tuesday.

Starmer told staff gathered at Labour’s HQ on Monday morning that they should get “100% behind” the party at the Makerfield byelection – where Labour will face a bitter fight with Reform UK. Nigel Farage’s party is expected to announce its candidate for the seat within days and on Monday released attack ads criticising Burnham’s frantic search for a byelection seat.

But Starmer also suggested he would fight any attempt to depose him by Burnham – after the mayor’s allies briefed that they hoped an orderly transition would take place by

“I do want to fight the next election,” Starmer said at an event on Monday. Asked if he would fight a leadership contest if another Labour MP gets enough support to mount a challenge, Starmer replied: “Well, we’re not at that position. But I’ve said, I don’t know how many times, that I’m not going to walk away. I feel very strongly that I must serve the people who voted me into office.”

Burnham – who is still officially yet to be picked as Labour’s candidate – said he wanted to turn the national spotlight on Makerfield and the north-west during his byelection campaign – saying his fight against Reform UK would centre on what could change for such places.

“I want to say sorry to the residents of the Makerfield constituency, for the circus that is about to arrive in town and some of the inconvenience they will experience as a result,” he said.

“But on the other hand, I want to say this to them as well: I hope you feel it’s a good thing as well, that the places that make up this constituency, long forgotten by national politics, finally are at the centre of the national debate. And for the places of this constituency again, you could read many of the similar places in yours.”

There is frustration among some in Labour that Streeting may have damaged Burnham and Labour’s efforts to retain Makerfield with his comments about the EU.

Number 10 said on Monday that Starmer had “red lines in the manifesto that we’re absolutely committed to” though stressed that Starmer was seeking a closer relationship with the EU.

However, experts in Brussels are warning that the UK would not be accepted back into the union on the same terms it once enjoyed. Georg Riekeles, a former adviser on the EU’s Brexit taskforce, said he expected member states would take “a very warm, welcoming” stance, but also a “hard-headed” one to a British membership application.

“There is a strategic need for the EU and the UK to work together, but I don’t think there would be an appetite for opening up new decades of British exceptionalism.”

Streeting, who has said he plans to campaign in Makerfield for Burnham, has been adamant he would also stand in a leadership contest.

Streeting’s next intervention is expected to come on Wednesday, when he hopes to use his first chance to speak in the Commons since his resignation to set out his ideas for Labour’s future.

One Labour source said that while Streeting’s comments on Brexit were most likely said on the basis they might hamper Burnham’s campaign, the view was not at all unexpected. “I always thought that Wes would speak about Brexit. He’s going to be campaigning from the right, so it’s probably only that and Gaza which help make him sound member-friendly,” they said.

Allies of the former health secretary denied that his comments on Brexit were intended to undermine Burnham, and that it was simply a case of someone who had been under collective responsibility for six years having a chance to freely say what he believed.

One said: “The point of the speech was to call for a proper debate of ideas, and what we stand for as a party. The reaction does reinforce Wes’s critique of a culture where debate gets shut down. This government is too afraid that someone might disagree with something we do, so we end up doing nothing.”