Margaret Cunneen faces a push to refer her to Parliament's powerful privilege and ethics committee after she demanded that MPs not provide details of secret phone taps involving her to journalists "or any other person".

The Crown prosecutor's lawyers delivered her demand in a letter to the chairman of the oversight committee on the Independent Commission Against Corruption, Liberal MP Damien Tudehope, on February 12.

It followed publication by Fairfax Media of excerpts of a transcript of Australian Crime Commission telephone intercepts that sparked the ICAC's aborted investigation of Ms Cunneen over allegations she tried to pervert the course of justice.

The excerpts show Ms Cunneen saying she had sent a message to her son's girlfriend, who had been drinking alcohol, "to start having chest pains" after a 2014 car accident, to avoid a breath test.