The author of the text warned they had suffered from mental health problems in the past and said being exposed publicly could push them over the edge and put their life at risk.
The appeal came after Mallard launched an inquiry into who leaked the expenses and promised to name and shame the person responsible.
Sources say the text was extensive, and was sent from a disposable phone.
It is understood police have also become involved and that National sought advice from mental health experts on how to deal with the text.
Mallard told the Herald this morning he will be making no comment on anything to do with the incident or the inquiry until the terms of reference are prepared: "end of story".
Newstalk ZB political editor Barry Soper has told Mike Hosking Breakfast that he understands it isn't a National Party MP.
"It could be anyone with a burner phone," Soper said.
"To me it's a storm in a teacup.
"Some National MPs are on their way to Wellington today, but not for a spill. It's a show of support for Bridges for when he speaks to the media this morning.
"It was quite a detailed text but, as I understand it at this stage, it isn't a National MP – but we'll have to wait and see."
RNZ said it had not seen the text, which is believed to have detailed a number of conversations and pieces of information from National caucus meetings over a period of weeks in an attempt to prove the author was a National MP.
Mallard has appointed former solicitor-general Michael Heron QC to investigate the leak, which showed Bridges had spent $113,000 in a three-month period on travel expenses and Crown limo costs.
Announcing the inquiry, Mallard said he believed someone had deliberately undermined the system, and the security of MPs was important.
"Unless they have incredible expertise they will be identified," he said.
Mallard believed the inquiry was a good use of taxpayers' money.
"Members of Parliament will not be able to do their jobs properly."
