Labour figures from across rival factions have begun circulating informal proposals for an “orderly transition” of power away from Keir Starmer, the Guardian understands.

MPs have shifted discussions from speculating about whether the prime minister could be removed to how – including timelines, potential triggers and the mechanics of forcing a leadership contest.

One Labour MP said: “There have been conversations about process. When the time comes, and if the numbers are there, a process will be found.”

With no formal mechanism to remove a sitting prime minister, attention has turned to how political pressure can be applied, with one senior party source warning poor local election results could trigger junior ministerial resignations, which would provide “cover for someone to come out from behind”.

Starmer has said he is determined to lead Labour into the next general election, but his allies acknowledge he does not have for ever to convince his party. A source speculated the prime minister had nine months to persuade MPs that he can turn things around.

Senior MPs said it was “overstated” to suggest any one faction was leading efforts to oust Starmer, with figures across the party instead engaging in a range of discussions.

There are competing claims about where the push is coming from. One figure claimed allies of Andy Burnham were driving calls for a longer transition to allow him to return to parliament before any leadership contest, while others suggested MPs aligned with Wes Streeting were behind efforts to accelerate the process.

Some MPs said rival camps were trying to shape the narrative around any move, reflecting a wider disagreement over timing and strategy across the parliamentary Labour party.

It is clear there is growing frustration among backbenchers that no potential successor had yet set out a clear direction, with one saying none of the names that had been touted were “actually putting a full-on manifesto forward”.

One MP said there was now near-universal concern across the PLP. “We need an orderly transition,” they said, adding: “Most people think it’s over for Starmer”.

A senior backbencher said morale across the party was at “rock bottom” with frustration spreading rapidly over the last week.

Starmer attempted to brush off mounting unrest inside his party and shut down questions about the future of his leadership in an interview with the Sunday Times. Asked directly if his prime ministership was over, Starmer told the newspaper: “No.”

Pushed on whether he could continue, he replied: “Of course. We didn’t wait 14 years to get elected, we didn’t change the Labour party, we didn’t do all that it entailed to win the election and the mandate for change, not to deliver on it.”

Asked if he would lead Labour into the next election, Starmer said: “Yes.”

A senior Labour MP said the “best chance of salvaging this” for Labour would be for Starmer to “name a timetable for departure”, with many expecting an orderly exit by conference to allow a successor to emerge.

It has been a bruising time for the prime minister since the Guardian revealed that he appointed Peter Mandelson as the US ambassador despite vetting officials recommending that he be denied security clearance. His handling of the row was called into question, including his swift decision to sack the Foreign Office chief Olly Robbins.

The prime minister’s allies denied claims of any wrongdoing over the appointment. Darren Jones, the chief secretary to the prime minister, told Sky News’s Sunday Morning with Trevor Phillips programme: “No wrongdoing by the prime minister has been proven in relation to Lord Mandelson’s appointment … The whole situation is regrettable.”

Supporters of Starmer have already planned his response to what are expected to be a difficult set of Scottish, Welsh and local English election results for Labour next month, which could once again throw his leadership into peril.

After the results roll in, they expect the prime minister’s initial reaction to be one of humility. “We know that voters are sending us a message: we need to acknowledge that we’ve heard it,” one source said. “Think of Obama’s ‘shellacking’ moment after the US midterms [in 2010] when the Dems took a heavy beating. Keir needs a similarly realistic and humble response.”

Downing Street also wants to “inject some hope” into the government’s narrative, one senior figure said, pointing to the king’s speech the following week as an opportunity to do so.

But they added: “We know that won’t be easy. We haven’t done a good enough job telling voters about all the big things we’ve done – and we’re too good at creating our own bad headlines. It’s not just down to Keir though, we all need to do better.”

Starmer hopes to bolster the political side of his No 10 operation – after the departures in recent months of his chief of staff Morgan McSweeney and director of strategy Paul Ovenden – with the recruitment of a heavyweight political strategist.

An insider said: “We know that people won’t be jumping to join us this side of the May elections, but we hope that once things settle down, it might be a more attractive option.”

Questions over Mandelson’s vetting will continue this week, with McSweeney due to appear before the foreign affairs select committee.