Andy Burnham needs to "build bridges" with the Starmerite minister he sacked during his reshuffle, Baroness Harriet Harman has told the Electoral Dysfunction podcast.
Baroness Ruth Davidson, a Tory, had slightly harsher words - branding Sir Keir Starmer's former lieutenant Darren Jones "bitter".
Mr Jones was chief secretary to the prime minister under Sir Keir before leaving government with the ex-PM.
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After Mr Burnham announced he was scrapping the government's digital ID scheme to spend hundreds of millions of pounds on cutting energy bills, Mr Jones posted on social media that the ID policy was not costed and so there was no money yet to save.
It crystallised criticism that Mr Burnham is pledging money to policies that sound popular, but he does not actually have any plans for funding them.
On the divisions within the Labour Party, Baroness Harman said: "[Burnham]'s also got to build bridges with the people for whom there has not been space in the cabinet, but are still members of parliament, have still got all the attributes and interests that they have, and that is perfectly possible.
"So as well as encouraging and supporting his current cabinet, with the work that they're going to take forward, he's got to talk to the people who are in the previous cabinet who've got expertise - and wisdom - they could feed in.
"He's got to be talking to them as well as talking to his current cabinet ministers. And if he does that, I think that he can build bridges."
She added that Mr Jones should be invited in by Mr Burnham and consulted as a "repository of a lot of wisdom and expertise".
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Baroness Davidson, however, was more cynical.
" Can I offer a different idea here, which is that Darren Jones knew exactly what he was doing when he sunk the first policy announcement of the new prime minister," she said.
Baroness Davidson added that Jones was "bitter" and "needs to be told that that's not how you behave".
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