Keir Starmer is not about to set a timetable for his departure from Downing Street, David Lammy, one of the prime minister’s closest cabinet allies, has said, calling on Labour to get beyond the “spectacular own goal” of repeated leadership speculation.

While allies of Starmer have suggested he is potentially willing to step aside if Andy Burnham wins next month’s Makerfield byelection and no other challenger emerges, Lammy insisted this was not being considered.

“There will be no timetable for departure,” Lammy, who is justice secretary and deputy prime minister, told Sky News.

“Let me be really clear – Keir Starmer remains the most resilient person I know in my life. I spoke to him twice yesterday. He has a strength of character, a fighting experience. There will be no timetables.

“What there is, is getting on with the business of government, [he is] really crystal clear about that. At the moment, there is no contest. What there is, is his determination to deliver for the people.”

Burnham is expected to be selected for the Makerfield contest after the sitting Labour MP, Josh Simons, stepped down specifically so the Greater Manchester mayor could try to return to Westminster. If he wins against an expected strong challenger from Reform UK, he is likely to challenge for the leadership.

Lammy said Burnham would “be a great addition to parliament”, and that he would go to the constituency, on the edge of Wigan in Greater Manchester, to campaign for him.

But he lamented the week of internal wrangling in which Simons stepped down and another potential challenger to Starmer, Wes Streeting, resigned as health secretary after seemingly failing to gather enough support from MPs to make a direct leadership bid.

“I’m not going to sugarcoat this, I thought that the Labour party over the last 10 days had a spectacular own goal after those local election results,” Lammy said.

“We now need to unite and pull together. We have a byelection to fight. We need to remember our responsibilities as a government. We are not in opposition. We have the levers of power.”

Reform UK are expected to campaign heavily on Makerfield over speculation that Labour might consider reversing Brexit, after Streeting used a speech on Saturday to say Britain’s long-term future lay in rejoining the union, something that was dismissed as “odd” by the culture secretary, Lisa Nandy.

Asked about the debate, Lammy said he was proud that when he had served as foreign secretary, he had resumed closer collaboration with EU nations, removed many trade barriers on food and agriculture products, and brought the UK back into the Erasmus student exchange scheme.

Asked about Streeting’s comments, Lammy said the government’s red lines, on not putting the UK back into the EU’s customs union or single market, let alone full membership, remained in place, adding: “I’m not going to make a commitment about the next election.”

Asked if he personally would like the UK to rejoin one day, he said: “I’m committed to collective responsibility, the manifesto we stood on. Wes Streeting has left the government. He can have a debate, he can comment. That is not my position.”