Jeremy Hunt, the health secretary, has retreated from accusations that his predecessor Andy Burnham was involved in an NHS cover-up, following a threat of legal action from Labour.

In a personal letter, Hunt said he never meant to suggest Burnham, now the shadow health secretary, was "personally involved in covering up poor care" and did not doubt his rival's integrity.

However, the cabinet minister did not delete a post on Twitter that refers to "shocking revelations on Andy Burnham's attempts to cover up failing hospitals". Hunt made the remarks after newspaper reports that Burnham's department interfered in the publication of a Care Quality Commission report about standards of care in Basildon hospital, which Burnham strongly denies.

Labour has demanded an apology or retration from Hunt, and threatened to sue over the claims made last week.

Hunt insisted his tweet refered to "political pressure on the Care Quality Commission over the publication process for reports of failure in hospital care".

But he also partially continued his attack on Burnham, saying he does "believe that the previous government, with you as the secretary of state for health, must acknowledge and take responsibility for these failures on your watch".

In the letter, Hunt said he was surprised that Burnham involved lawyers, saying the right place for elected politicians to debate was in parliament or the media, not the courts.

Hunt moved to calm down the row in the middle of a reshuffle that saw Burnham keep his job as shadow health secretary, despite speculation he would be moved from the role.