Labour should not be “doomscrolling” through leaders like the Conservatives, the communities secretary, Steve Reed, has said, urging MPs not to move against Keir Starmer after the May elections.

MPs who fear Starmer cannot lead the party into the next general election because of his unpopularity are understood to have been discussing whether to lay out a timetable for his departure to present to the prime minister.

Starmer could also potentially face a direct leadership challenge, though would-be candidates, including Wes Streeting and Angela Rayner, are said to be unlikely to move first against him.

MPs told the Guardian they were sceptical about the idea of a letter urging the prime minister to set out a timetable for departure – though some hope a longer timeframe would benefit the mayor of Greater Manchester, Andy Burnham, giving him time to return to Westminster.

Reed said the majority of MPs and local leaders did not want to unleash the chaos of a leadership challenge. “I speak to a lot of my fellow MPs, of course I do, all the time, but also council leaders, and they’re sick and tired of all this psychodrama,” he said.

“They want us to focus as a party on what we need to do to get our vote out this coming Thursday. There are really important issues about who runs our councils, whether we can build the social housing that this country needs, whether we can improve the public services that people use,” he told Times Radio.

“The whole notion that we would copy the Conservatives and go doomscrolling through leaders in a way that means the government is completely incapable of dealing with the things that matter to most of the British public is absolute nonsense, and I’m not going to engage in it, and most of our MPs would not engage in that either.”

Labour could lose more than 1,500 council seats across England, and face a struggle for second place in Scotland and the prospect of losing Wales after a century of domination, leaving thousands of angry local politicians who see themselves as victims of the government’s unpopularity.

However, the vast majority of cabinet ministers are thought to be unwilling to move against Starmer.

“There’s a complacency on the backbenches,” one cabinet minister said, “particularly among new MPs, that any Labour leader could’ve won the [last] election, so they don’t give Keir credit for it, and think they can plot and say we should replace him. They’re wrong.”

A second cabinet minister said the 2024 intake, unlike more seasoned backbenchers, had only ever experienced the “upside of the cycle” – winning elections rather than facing a midterm thrashing – and they felt anxious as a result.

A third said that the MPs pushing for Starmer to set out a timetable were allies of Burnham. “It’s really about pressure to let Andy back into the party.”