There have been calls to divide the roles of the post to avoid a perception of a conflict of interest, as the Lord Advocate leads Scotland's prosecution service as well as acts as the most senior legal adviser to the Scottish Government.
Concerns about the dual remit were raised by MSPs in the wake of the Holyrood inquiry into the government’s handling of sexual harassment allegations against former first minister Alex Salmond and the malicious prosecution of former Rangers FC directors.
While justice is devolved, the remit of the Lord Advocate is laid out in the Scotland Act, which means the legislation would need to be amended by Westminster if Holyrood wanted the role to change.
According to a report in The Times, East Lothian Alba MP Kenny MacAskill had written to Mr Johnson asking if he would consider changing the law.
A reply sent on Mr Johnson's behalf by Alister Jack, the Scottish Secretary, said: “Whilst the UK Government has the power to make this change, in practice we would want to ensure the Scottish Government had put their proposals to the Scottish Parliament for scrutiny.
“It would then be for the Scottish Government to make representation to the UK Government.”
Mr Jack said no steps had yet been taken to change the law.
First Minister Nicola Sturgeon has said there was a "strong, prima facie case" for the separation of powers, repeating her election pledge to consult on whether changes should be made.
James Wolffe, who stepped down from the role after the election, was replaced this week by Dorothy Bain QC, who has said she would give any consultation on the role “all appropriate assistance”.
Mr MacAskill, a former Scottish justice secretary said the Scottish Government could adopt immediate changes to begin the separation by declining to approach Ms Bain for legal advice. Such a move would be “awkward, but not insurmountable”, he said.
"This has been an issue long before the election,” he said.
"Surely engaging to be able to act in the wake of it would have been sensible. There are complexities for sure so get on with it now ... and get it sorted."
According to a survey by Scottish Legal News in March, 81.4 per cent of the 350 lawyers questioned want the role divided into equivalents of an attorney general and a director of public prosecutions.
The Scottish Government said: "We intend to publish a consultation on the dual functions of the law officers in due course. It is vital that the systems of criminal prosecution and investigation of deaths, as well as the role of the law officers in government, continue to enjoy the full confidence of the public.
"The First Minister has also been clear that, as this is an important constitutional matter, it is vital to take the time to get the consultation right."