Angela Rayner has warned that Labour faces its "last chance" after heavy election losses this week, as she backed Andy Burnham to return to Westminster.
In a statement following Labour's disastrous performance at the polls, the former deputy prime minister said the party had been wrong to block the Greater Manchester mayor from standing as an MP earlier this year.
Rayner, seen as a potential contender to replace Sir Keir, called on him to "meet the moment" with bolder action to make people feel better off.
She did not launch a leadership challenge herself, but her intervention will add to pressure on the embattled prime minister, who will try to shore up his premiership in a speech on Monday.
In her first public comments since Thursday's elections, Rayner called for Sir Keir to take more action to tackle inequality and "squeezed living standards".
In her 1,000 word statement, she called for Labour to offer regional mayors more economic powers, raise the minimum wage, and be "unafraid to promote new forms of public, community and cooperative ownership across the board".
"We are in danger of becoming a party of the well-off, not working people," she added.
"What we are doing isn't working, and it needs to change. This may be our last chance."
Alongside Health Secretary Wes Streeting, Rayner and Burnham are thought to be the main potential challengers to Sir Keir's leadership.
In her statement, Rayner did not endorse the Greater Manchester mayor as a leadership candidate, but said it had been a "mistake" for the party to block him from standing at February's by-election in Gorton and Denton.
"We must show we understand the scale of change the moment calls for - that means bringing our best players into Parliament," she said.
She added that this involved "embracing the type of agenda that has been successful at a local level, rather than reaching back to an agenda and politics that has failed people".
Labour lost almost 1,500 councillors in local elections across England amid a surge for Nigel Farage's Reform UK, with the Greens also eating into the party's support in London and other urban areas.
It was also kicked out of power in Wales, where it has enjoyed political dominance for a century, and returned just 17 of the 129 seats in the Scottish Parliament, its worst ever result at a Holyrood election.
Sir Keir now faces another key hurdle on Monday, when backbench MP Catherine West says she will seek to trigger a leadership contest against him if she is "still dissatisfied" after his reset speech.
West, a former junior minister, is not bidding to be the next Labour leader herself, but by triggering a leadership she could flush out heavyweight contenders she thinks could do a better job than Sir Keir.
More than 30 Labour MPs have publicly called for the prime minister to resign or to set out a timetable for his departure, including former minister Josh Simons, who said the prime minister had "lost the country".
Education Secretary Bridget Phillipson earlier warned MPs against triggering a contest, and backed Sir Keir to lead Labour into the next general election.
She conceded that voters did not feel the Labour government had "delivered" on the change it promised, but insisted that the public did not want to see Labour "fighting amongst ourselves" in a leadership contest.
Burnham cannot enter a contest without first becoming an MP, under the party's rules for leadership contests. The mayor's supporters are hoping a leadership contest can be delayed until he is able to return to Parliament.
Rayner resigned as Sir Keir's deputy prime minister last September after it emerged she had failed to pay the correct amount of stamp duty on a £800,000 flat in Hove.
It has been widely reported that she is still waiting for the result of an HMRC investigation into the purchase, which could complicate any immediate leadership campaign.
