Cabinet ministers have warned mutinous Labour MPs that any attempt to oust Keir Starmer after a potentially disastrous set of election results this week would unleash chaos for the party that would not be easily overcome.

However, several told the Guardian that even with the prime minister’s determination to stay in Downing Street after Thursday’s vote, the mood on the backbenches was febrile and events could yet spiral out of control.

They also admitted that – while they would discourage any coup against Starmer now, they did not expect him to lead the party into the next election. “When your personal brand is so poor, it is seldom retrievable,” one added.

Labour faces losing more than 1,500 council seats across England, 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.

Before they headed to their constituencies last week, MPs were gripped by speculation over Starmer’s future, with Angela Rayner, Wes Streeting and Andy Burnham – despite not being eligible – seen as likely successors.

Some have argued Starmer should set out a timetable for his departure – and have suggested that a group of cabinet ministers might be prepared to tell the prime minister that his time was in up if the results are as bad as predicted.

The Guardian understands, however, that the appetite inside the cabinet for a leadership contest is severely limited, even among those ministers who believe that he will step down before the 2029 general election.

“We have a role to play and we’ll certainly not want chaos,” one said. “That’s not in anybody’s interests.”

Another indicated there was no group within the cabinet that was planning to move collectively, while a third added: “I don’t want new leaders, plots, pacts, talk of orderly transitions which shut out the public. Will there be cabinet resignations or a move against Keir? There could be, but I won’t be part of it.”

Several ministers warned of the danger of unintended consequences. “Those of us who are sane don’t really want a leadership contest or a timetable for Keir’s departure that undermines the party’s position, but we recognise that when the mood is febrile things can kick off,” one said.

“We wouldn’t be thanked for picking our own leader three years out from the next election. It’s not that things are perfect, it’s just that it’s premature.”

Neither Rayner, the former deputy PM, nor Streeting, the health secretary, are thought likely to move first – with allies suggesting they would only enter a contest if it were triggered by somebody else. In addition, Rayner still has to resolve her tax affairs.

Members of Labour’s national executive committee – which blocked Burnham from standing in the Gorton and Denton byelection in February – do not believe there is a route for him through it back to parliament, despite reports at the weekend.

Ed Miliband, the climate secretary, is said by colleagues to be more likely to try to act as kingmaker for Burnham, rather than to go for the top job himself. One friend said he was determined to prevent Streeting from becoming leader.

Downing Street has made clear that Starmer would fight any attempt to oust him, with allies of the prime minister saying any putative rivals should think carefully about both the instability any challenge would cause, particularly at a time of conflict and with difficult economic headwinds.

They also played down the prospect of any reshuffle – regarded by some as potentially even more destabilising – suggesting he would only do one if ministers resigned following the election results and he had to fill the gap.

Writing in the Observer at the weekend, Starmer said: “We have a choice. We could sink into the politics of grievance and division. Or we could rise to this moment – together – in a national effort that matches the scale of the threats and turbulence we face.

“When the nation rallied together to deal with Covid, the last government could have channelled that spirit to build a better nation. But instead, they descended into political infighting and let the country slump back to the old status quo. Not this time.”