The UK culture secretary has dismissed calls for Britain to re-enter the European Union as “odd”, a day after her former cabinet colleague Wes Streeting said the country’s future lies back in the bloc.

Lisa Nandy criticised the former health secretary on Sunday, saying his comments over the weekend risked reopening a debate that was settled with the Brexit referendum 10 years ago.

With the issue now likely to feature prominently in the Makerfield byelection and the Labour leadership contest that is expected to follow, Nandy insisted the prime minister, Keir Starmer, was right to have ruled out rejoining the EU.

She told the BBC’s Laura Kuenssberg: “I actually think this is just a bit odd. I listened to what Wes had to say very carefully yesterday, and I know that he’s got a strong view about this, and always has had, that we shouldn’t have left the European Union.

“Frankly, that’s one that I share. I campaigned for remain, I think it [Brexit] was a mistake, and I think the Brexit deal has been a real problem for us. But I don’t really understand why the sudden focus on Europe.”

She added: “We’re already, as a government, trying to repair in a pragmatic way the needless damage that was done by that poor Brexit deal to people’s living standards in towns like mine, without reopening the circular arguments that we ended up in as a country.”

Streeting’s remarks at a speech in London have put the issue of Britain’s relationship with the EU high on the agenda in the Makerfield byelection, where Greater Manchester’s Labour mayor, Andy Burnham, intends to run on a promise to challenge Starmer for the party leadership.

Streeting, who this week resigned from the cabinet in protest against Starmer’s leadership, said on Saturday: “In 2026, the British people increasingly see that in a dangerous world we must club together, both to rebuild our economy and trade, and improve our defence against the shared threats from Russian aggression and America First.

“The biggest economic opportunity we have is on our doorstep. We need a new special relationship with the EU, because Britain’s future lies with Europe – and one day back in the European Union.”

He added that he intended to stand if a leadership contest was started – though it remains unclear whether he has enough parliamentary support to do so.

The byelection has not yet been formally called and nor has Burnham been chosen as Labour’s candidate, yet his policy positions are already being carefully picked over as signs for what he might do if he goes on to become prime minister.

Burnham told ITV News on Saturday that rejoining the EU would not be a focus in his campaign. “In the long term there is a case for that, but I’m not advocating that in this byelection,” he said. “In fact, what I am saying is focus now domestically. Britain has got to focus very much on the here and now and the issues that are affecting people.”

But Josh Simons, the Labour MP who is vacating the Makerfield seat to make way for his ally Burnham, suggested that nationalisation of utilities could be an important part of his pitch to voters.

“Energy, water, social housing – those things that are the basics of our lives that we all depend on – have gotten so expensive,” he told the BBC on Sunday.

“And one of the reasons why they’ve gotten so expensive – not the only, but one of the reasons why they’ve gotten so expensive – is that we privatise a lot of them, and often the bills that we pay go to the shareholders.”

Meanwhile, allies of Starmer say the prime minister has not given up hope of remaining in office, insisting he will enter the race if a leadership election is called.

Asked on Sunday whether she thought Starmer would run in a leadership contest, Nandy said: “He said he will … I haven’t spoken to him this weekend, but I’ve spoken to him several times over the last week, and obviously he’s shown before that he’s up for a fight.”