"You can't break the law of the land," the source said.
Mr Brown is also losing patience with both sides in the long-running port dispute following the breakdown of mediation talks yesterday.
He said it was time for the two sides to put aside entrenched differences and consider Auckland's best interests.
"The majority of Aucklanders would agree with me that the dispute has gone on way too long," the mayor said.
He supported a proposal by the port company for the parties to use the Employment Relations Authority's facilitation process. The Maritime Union is considering the option.
Ports of Auckland chief executive Tony Gibson has acknowledged in a letter to the union that the company was behind the leak to blogger Cameron Slater, who published it on his Whale Oil website, but did not say who was responsible for the leak.
He said the release of the information was intended purely as a response to Mr Walker's commenting negatively and unfairly in public about the company.
Mr Gibson said: "Upon reflection, we accept the distress this may have caused him, and for that we are very sorry."
Yesterday, the company refused to answer questions from the Herald about whether any senior staff or board members authorised the leak.
Nor would the company say if any action had been taken against those responsible.
The Council of Trade Unions has asked the Privacy Commissioner to investigate the matter and determine who leaked the information.
CTU president Helen Kelly said the union wanted a proper apology and compensation for Mr Walker, saying the comments and responses on the Whale Oil website were disgraceful - which was what the port company wanted.
"They wanted to belittle him [Mr Walker] in the eyes of the community," the union leader said.
Last month, striking port workers said they felt violated by the release of highly personal information - including bereavement leave details - to Mr Slater's blog site.
Information he published on Whale Oil includes a list of 106 leave days - under five categories - Mr Walker, a crane driver, took from the time his wife was diagnosed with a terminal illness in 2007 until she died the following year.
Yesterday, Mr Slater - son of former National Party president John Slater - reiterated that he received the information about Mr Walker from an anonymous "tip line" and had no idea of its source.
"The port company has admitted it came from them. I didn't know it came from them."
Mr Slater said he had been sent "a whole lot more information" of a deeply personal nature that he had not published because it was not relevant to the port workers' criticism of the company.
Councillor Mike Lee, who as chairman of the old Auckland Regional Council oversaw the port company, said the leak was an excessive and vindictive thing to do.