It comes after former Foreign Office chief Sir Olly Robbins said there was a "dismissive approach" on vetting from No 10 and an "atmosphere of pressure" to get Lord Mandelson's appointment as UK ambassador to the US over the line.
The Labour leader is expected to be grilled at Prime Minister's Questions after Sir Olly defended himself in front of the Foreign Affairs Committee on Tuesday.
The senior civil servant chief was sacked by the prime minister last week after it emerged he did not tell him that Lord Mandelson was cleared to take up the top diplomatic post despite security vetting officials recommending against it.
Sir Keir told the Commons on Monday he had challenged Sir Olly over why he went against the recommendation of UK Security Vetting (UKSV) and did not accept his explanation.
But Sir Olly said it was normal not to share the findings of the vetting process and described feeling under pressure to clear the peer for the role.
When Sir Olly took over at the helm of the Foreign Office in January last year, Lord Mandelson had already gone through the Cabinet Office's "due diligence" process, the King and the US had given him approval, and he was already being granted access to "highly classified briefings" on a case-by-case basis – without his security clearance being confirmed, he said.
The Whitehall veteran said this resulted in a "dismissive approach" to developed vetting from Downing Street for the remainder of the process but insisted it was completed to the normal standard "despite this atmosphere of pressure".
No 10 has rejected this, saying there is a distinction between "the idea of pressure" and "being kept informed about the process and the progress of the appointment".
Meanwhile, Dame Emily Thornberry suggested she could bring Sir Keir's former chief of staff, Morgan McSweeney, before the Foreign Affairs Committee.
The senior Labour MP, who chairs the committee, was asked by LBC if she would ask the long-time Mandelson ally to appear and responded: "I am going to invite some other witnesses. It would probably be best if they heard that first from me rather than from you."
Tory leader Kemi Badenoch and the SNP have called for Labour to hold a vote of no confidence in Sir Keir.
And a leak inquiry is under way over the story being given to media, Chief Secretary to the Prime Minister Darren Jones confirmed.
Sir Olly had called it a "grievous breach of national security" that information was given to The Guardian after the Cabinet Office briefed Number 10 on the matter.
Chancellor Rachel Reeves backed the decision to sack Sir Olly, telling an event that whether Lord Mandelson had passed vetting or not was "crucial information" that the mandarin should have told the prime minister.
She insisted there is "no Labour leadership contest" when asked about the possibility of challengers to Sir Keir's authority.
At the same event, former deputy prime minister Angela Rayner said Labour should take "bold action" to show people how they are tackling the crisis surrounding the Iran war.
Though media reports have suggested Ms Rayner recently met Greater Manchester Mayor Andy Burnham amid rumblings of a leadership challenge, sources close to Ms Rayner suggested her speech signalled she was open to a return to serving as a minister in Sir Keir's Government.
Energy Secretary Ed Miliband earlier said he had feared Lord Mandelson's appointment would "blow up" and had spoken to then-foreign secretary David Lammy about his concerns.
But the former Labour leader said he did not think Sir Keir should resign over his decision to appoint Lord Mandelson to the post.
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