Andy Burnham has backed stark criticism of the direction of Keir Starmer’s government by Angela Rayner after she said the very survival of the Labour party was at stake.

Rayner, the former deputy prime minister and an influential backbencher, used a speech on Tuesday night to warn that the prime minister “cannot go through the motions” in the face of declining support.

Burnham, the mayor of Greater Manchester, who is regarded as a rival to Rayner in any potential leadership contest to succeed Starmer, said on Wednesday: “I certainly know where she is coming from and the party would always do well to listen to what Angela has got to say.”

In comments that will be seen alongside Rayner’s intervention as an effort to shift the party to the left as it faces the prospect of heavy losses in local elections in May, Burnham said the frustrations she was reflecting had also been heard at the recent Gorton and Denton byelection.

Burnham had wanted to stand in that election for a previously safe Labour seat which ended up being won by the Greens but was blocked by the Labour leadership.

Burnham was speaking on the morning after Rayner used a speech at an event in London organised by Mainstream, a Labour campaign group linked to Burnham, to make her most significant intervention since resigning last year from Starmer’s government.

Rayner, who has continued to command influence from the backbenches, also said it was “un-British” to move the goalposts on indefinite leave to remain, putting her at odds with the government’s key immigration proposal of increasing the standard qualifying period for permanent residence from five to 10 years.

“It is down to us to rebuild this nation and this party – the question is are we up for this fight? I know we in this room are,” Rayner told the gathering in London on Tuesday evening.

“As a party, and as a movement, we cannot hide, we cannot just go through the motions in the face of decline. There’s no safe ground and we’re running out of time. The change that people wanted so desperately needs to be seen, it needs to be felt. And we have to show that it is a Labour government that will deliver it.”

Speaking on the Today programme on BBC Radio 4, Burnham voiced his own criticism of the government’s approach to immigration, saying that Rayner had been echoing “moral questions”. But he added that net migration had started to come down, saying: “I do think the government has a story to tell here and needs to tell it more effectively.”

In the first response to Rayner’s remarks from a government figure, the Cabinet Office minister Nick Thomas-Symonds said Starmer’s administration shared her “impatience with the pace of change”.

Asked if he agreed with Rayner that Labour was going “through the motions in the face of decline”, he told Sky News: “I think where I would agree, and I think everybody across government would agree, is sharing an impatience with the pace of change, and that applies to every single one of us.

“And I get the sense, I haven’t read the full context of Angela’s remarks, but I get the sense that that frustration is actually what is running through her remarks. It absolutely runs through every government minister as well.”

He defended Shabana Mahmood’s immigration plans as “balancing up fairness, but also security at our borders” when asked about Rayner’s criticism of them. He declined repeatedly to say whether he thought Rayner would make a good party leader, stressing that there was “no vacancy” for the role.