Alex Salmond has called for a probe into whether Nicola Sturgeon breached the ministerial code to be extended.
The first minister referred herself for investigation by an independent panel over whether she broke any rules when complaints against Mr Salmond emerged.
Ms Sturgeon insisted that she acted "appropriately and in good faith".
However Mr Salmond has said the remit of the current inquiry is a "straw man" designed to be knocked down, and said other issues could be looked into.
The former first minister has written to standards panel member James Hamilton to ask if he will investigate whether Ms Sturgeon had breached the ministerial code by misleading parliament or failing to act on legal advice.
The ministerial code states that "ministers who knowingly mislead the parliament will be expected to offer their resignation".
The Scottish government said the remit of the investigation was "well established" already.
The probe is linked to the Holyrood inquiry into the Scottish government's botched handling of harassment complaints against Mr Salmond.
Ms Sturgeon had meetings and phone calls with her predecessor at the time the claims were being investigated by her government.
After the government admitted in court that its investigation had been unlawful due to a procedural flaw - and agreed to pay Mr Salmond £500,000 in legal expenses - Ms Sturgeon ordered an independent investigation into her own conduct.
She said she had not interfered in the investigation, and was confident that she had "acted appropriately and in good faith throughout, and in compliance with the ministerial code".
Mr Hamilton - a former director of public prosecutions in Ireland - was tasked by Deputy First Minister John Swinney with examining Ms Sturgeon's role, if any, in the investigation into Mr Salmond.
The inquiry was put on hold for the duration of Mr Salmond's criminal trial - which saw him acquitted of 13 counts of sexual assault - but has now resumed.
Central to its remit is to determine whether Ms Sturgeon attempted to "influence the conduct of the investigation" into Mr Salmond after she discussed it with him.
However Mr Salmond - who was first minister when the independent panel on the ministerial code was set up, and was investigated and cleared of breaching it himself on five occasions - suggested this was too narrow an aim for the inquiry.
He said "it might even be suspected that this remit has been set up as a straw man to knock down", and suggested a range of further points which might be looked into.
These include whether Ms Sturgeon failed to act on legal advice, failed to ensure civil servants "give truthful information to parliament" and whether or not she herself misled parliament.
They were once close political allies, but Alex Salmond and Nicola Sturgeon have not spoken since those now-famous meetings over the internal investigation in 2018.
And now the breach between these two heavyweights of Scottish politics is becoming increasingly public - and personal.
Having faced several of these probes himself, Mr Salmond knows all about the ministerial code.
He also knows that the ultimate sanction for misleading parliament - a rule he specifically highlighted to Mr Hamilton - is that the minister concerned lose their job.
It is hard to read this as anything other than a direct challenge from him to Ms Sturgeon's position as first minister.
Ms Sturgeon has repeatedly come under pressure from opposition parties about her dealings with Mr Salmond, and has insisted that she has "nothing to hide in all this".
The Conservatives have already accused her of having misled parliament over when she learned about the internal complaints against Mr Salmond.
The first minister said she first learned of the investigation from Mr Salmond in a meeting at her Glasgow home on 2 April 2018 - but later said she had met with his former chief of staff, Geoff Aberdein, at Holyrood on 29 March.
Ms Sturgeon said she "forgot" about this meeting, because the encounter with Mr Salmond days later had left her "shocked and upset" and was "seared on [her] memory".
She has also hit out directly at her predecessor, saying in a Sky TV interview that he may be angry with her for refusing to "collude" with him to make the internal complaints "go away".
She said: "I understand that for Alex Salmond it is probably better for the focus of this to be on people wanting to believe there was a big conspiracy against him, as opposed to the focus being on his conduct.
"This is age-old here, that a man is accused of misconduct against women and often it's a woman that ends up sitting answering for them."