The Police Investigations and Review Commissioner (Pirc) is carrying out an investigation into the death of the 31-year-old who died in May following a police incident during which a female officer was injured.
Lord Advocate Frank Mulholland urged all those with an interest in the death of Mr Bayoh not to engage in “speculation and a running commentary”.
Mr Mulholland called for Pirc and the Crown Office to be allowed to “get on with their job” amid intense media interest in the case.
The intervention follows a public row between Aamer Anwar, the lawyer representing Mr Bayoh’s family, and former justice secretary Kenny MacAskill.
Yesterday, Mr Bayoh’s sisters, Kadijartu Johnson, Adama Jalloh and Kosna Bayoh, called on First Minister Nicola Sturgeon to investigate Mr MacAskill for “bullying”, claiming his comments had been “brutal and barbaric”.
Earlier this week, Mr MacAskill was accused of pre-judging the Pirc investigation after he wrote an article for a police journal in which he said it was likely to reveal Mr Bayoh’s death had been a “tragic accident”.
Mr Mulholland said both Pirc and the Crown were “well aware of all the evidence, the lines of enquiry and the issues surrounding this case”.
“The Crown and Pirc are not influenced by comments made in the media and that is how it should be,” he said.
“However, speculation and a running commentary on the investigation can be upsetting to the family of Sheku Bayoh as well as the families of the officers involved.
“A decision will be taken at the end of this extremely complex investigation as to whether or not criminal proceedings should be raised.”
He continued: “An inquiry will also be held at which all the relevant evidence will be heard, open to the public and the media, and it is right that it is this forum where the evidence will be rigorously tested and judicially assessed. Pirc, and the Crown, should be allowed to get on with their job.”
Speaking yesterday, Mr Anwar said: “The Bayohs are grateful for [the] intervention by the Lord Advocate and hope that those who should know better will stop declaring publicly that there will be no criminal proceedings when there is an ongoing investigation.”
Scottish Labour’s justice spokesman, Graeme Pearson, said: “There are processes in place to investigate and report this matter and we should not seek to prejudge outcomes in the absence of evidence one way or another. If the former cabinet secretary for justice Kenny MacAskill has privileged knowledge to support his media interventions on this case he should share it with Pirc and the Crown Office.”
A Pirc spokesman said: “The commissioner is committed to delivering a thorough, impartial and independent investigation and a final report of our findings will be sent to the lord advocate in due course.”
Mr MacAskill was unavailable for comment.