Ms Sturgeon accepted that the change should be considered when challenged over the resignation of chief constable Phil Gormley at First Minister’s Questions.
Mr Gormley’s departure dominated the weekly joust at Holyrood with Richard Leonard renewing Labour’s calls for Justice Secretary Michael Matheson to quit over his handling of the chief constable’s position.
Mr Matheson has faced accusations of political interference in policing after he met with the then chair of the Scottish Police Authority (SPA) Andrew Flanagan to discuss the watchdog’s decision to allow the then chief constable to return to work.
Following the conversation in November, the SPA reversed its decision and Mr Gormley remained on special leave until he resigned this week.
At First Minister’s Questions, Scottish Conservative leader Ruth Davidson asked Ms Sturgeon if she thought Police Scotland was being well managed, given that there had been the resignations of two chief constables and a Justice Secretary “pulling the strings when it suits him”.
Ms Davidson went on to criticise the fact that the SPA chair is a ministerial appointee.
The Tory leader said: “The head of the Scottish Police Authority is supposed to be independent of government yet it is Justice Secretary that appoints them. And as this affair has shown us that same Justice Secretary can pull the head of the Scottish Police Authority into a room and make him change his mind. Does the First Minister think that sounds like true independence to her?”
Ms Sturgeon replied saying the Justice Secretary had acted “entirely appropriately” when he discussed Mr Gormley’s return to work with the then SPA.
She added that she was “open” to changing the way SPA chairs are appointed, but warned that it would require changing the law.
The First Minister said: “We are open to looking in the future at how further changes can be made. But we have to be frank in telling parliament that substantial changes to that appointment process would require primary legislation. But we are open to discussing that and I am sure these are debates that will be taken forward in the months ahead.”
Ms Davidson pointed out the Information Commissioner was selected by a cross party panel, approved by the parliament and was therefore independent of government.
“That is exactly what we need from a police authority chair as well,” Ms Davidson said. “The First Minister is correct to say that five months ago, every single party in this chamber, bar the SNP, signed up for parliament to be in charge of appointing the SPA chair – to take it out of the hands of ministers and like the appointment of the information commissioner to put it in the hands of this whole chamber.
“So the First Minister stands here again, five months after she stood here before and says she can’t go further because it requires changing the law. Guess what First Minister. This is a parliament and changing the law is what we do.”
Ms Sturgeon replied: “Of course we can consider whether legislative change would be appropriate. Can I suggest that it is proper to consider that fully and robustly and why should we take time? Because we have a new chair of the Scottish Police Authority in place. She is at the start of her term of office. I think she is doing an excellent job and I think we should get behind her. Yes, I think we should consider in the fullness of time before we come to appoint a new chair whether change is necessary.”