Keir Starmer will not lead his party into the next general election, Labour-supporting unions have predicted, in an intervention that threatens to further destabilise the prime minister after a damaging few days.

The 11 Labour-affiliated unions – which include Unite, Unison and the GMB – are expected to issue a joint statement on Wednesday saying “at some stage” the party will have to put a plan in place to elect a new leader.

At a private meeting on Tuesday, the unions were divided over whether to call for Starmer to set out a timetable for his departure, with one source telling the Guardian there had been a “big fight” among union officials.

However, they are understood to have agreed to issue a statement saying they expect there to be a change of leadership, despite GMB and Community arguing it was not in the unions’ best interests to get involved in leadership wrangling.

In a leaked copy of the statement, the unions said it was clear to them that Labour “cannot continue on its current path”, and despite some progress it was not doing enough to deliver the change people voted for at the last election.

They urged the party leadership to focus on the “fundamental change of direction on economic policy and political strategy” they said working people needed, rather than the “personalities and unfolding political drama” at Westminster.

Starmer was increasingly confident that he had seen off the immediate threat to his job on Tuesday after a challenge from Wes Streeting failed to materialise despite several of the health secretary’s allies quitting the government.

But the prime minister’s fragile authority has been weakened by the resignation of four ministers – three of them close allies of Streeting – in what appeared to be an orchestrated move. More than 90 Labour MPs have also called for him to go since the weekend.

Starmer, who told his cabinet he would fight on as prime minister after a turbulent few days, was hoping that his second king’s speech on Wednesday would be another reset moment for the government to help unite his deeply divided party.

While he appears to have survived for now, even his most loyal ministers have privately acknowledged that he is unlikely to take Labour into the next election, unless he can dramatically turn round his and the government’s fortunes.

In their draft statement, which is due to be released on Wednesday, the union general secretaries wrote: “Labour’s affiliated unions have been clear that Labour cannot continue on its current path.

“Whilst we recognise progress has been made, such as aspects of the Employment Rights Act and the increase in the minimum wage, the results at the election last week were devastating.

“Labour is not doing enough to deliver the change that working people voted for at the general election. Our focus is on the fundamental change of direction on economic policy and political strategy that unions have been clear is needed, and not on the personalities and unfolding political drama in Westminster.”

But they added: “It’s clear that the prime minister will not lead Labour into the next election, and at some stage a plan will have to be put in place for the election of a new Leader.

“This is a point where the future of the party we founded will be debated and determined – and we are working closely as unions to shape a shared vision on policy, political strategy and economic policy that will reorient Labour back to working people, so Labour do what it was elected to do: govern in the interests of workers.”

Union officials had been frustrated when Downing Street postponed a meeting of the Trade Union and Labour Party Liaison Organisation [TULO] due on Tuesday, but agreed to meet for talks anyway.

The Labour leadership has had an increasingly tense relationship with the unions – which help to fund the party – since coming to power, despite big wins for unions on issues including workers’ rights and the minimum wage.

Some union leaders have urged Starmer to quit, with Unite’s Sharon Graham saying the “writing is on the wall” for the prime minister after last week’s election disaster. Others have urged the party to focus on its plan to change the country rather than arguing about the leadership.