Miller has apologised to MPs after she was ordered to repay £5800 for over-claiming mortgage expenses.
But Labour said she and the PM should admit "what they knew" about a call made to a Telegraph reporter by Jo Hindley, Miller's special adviser, in which she referred to the Leveson press inquiry.
The comment has been interpreted as a threat to the newspaper to drop its coverage.
The call for clarity came as a tape of the telephone conversation was made public.
However, Conservative Party chairman Grant Shapps said he did not believe that Hindley's reference to Leveson amounted to a breach of the special advisers' code of conduct.