The remarks follow comments by Ian Oliver, chief constable of Grampian Police, who on Sunday described the behaviour of Henry McLeish, the Scottish home affairs minister, as "deplorable" and that comments he had made were "intemperate".
He was responding to Mr McLeish's critical remarks about Dr Oliver's decision to attend a conference in Taiwan as a controversial report was released.
Mr McLeish said last week he was unhappy with Grampian's internal inquiry into the way it handled the investigation into the murder of Scott Simpson, nine, who was strangled by a paedophile.
The war of words continued yesterday when Mr McLeish said there was nothing to be gained "by seeking to indulge Dr Oliver by continuing this public slanging".
In an interview with BBC Scotland, he added: "The key issue is this - a young boy, Scott Simpson, was tragically murdered by Steven Leisk. We want to get to the bottom of what the police investigation involved itself in.
"We want to learn the lessons and I would suggest to the chief constable of Grampian that he now concentrates his energies in ensuring that we get a speedy, successful outcome to that inquiry."
A Grampian Police spokeswoman said Dr Oliver did not wish to comment further.