Sources close to Mr Burnham told The Mail on Sunday that Miss Harman advised him that the best way to end the controversy was to say sorry to director of Liberty Shami Chakrabarti.
Deputy leader Harriet Harman told Andy Burnham to apologise
Culture Secretary Andy Burnham: Stopped short of a full apology
He wrote to Miss Chakrabarti on Friday, saying he had meant no offence - but stopped short of a full apology.
Mr Burnham had been attacked by MPs from all parties for claiming Miss Chakrabarti shared 'late-night, hand-wringing, heart-melting phone calls' with Mr Davis in their joint campaign against the Government's plan to detain terror suspects for up to 42 days without trial.
Publicly, Miss Harman defended Mr Burnham's response, but The Mail on Sunday understands that, privately, she took a much tougher stance.
Former Shadow Home Secretary Mr Davis is forcing a by-election in his Haltemprice and Howden seat in East Yorkshire.
He and Miss Chakrabarti, both happily married to other people, have denied any improper relationship.
David Davis and Shami Chakrabarti: They have denied any improper relationship
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