Mr MacAskill said that Mr Murrell, who is the husband of First Minister Nicola Sturgeon, should quit his role at the top of the party to avoid the ‘suggestion’ that he was looking out for his spouse.
The outspoken comments from the former Edinburgh Eastern MSP will be viewed as a direct challenge to the First Minister, who has come under pressure from within the SNP over how she handled the allegations against Alex Salmond.
Ms Sturgeon is also facing a parliamentary probe into discussions with Mr Salmond while an investigation into complaints of sexual harassment – which the former First Minister denies – was ongoing.
There has long been a suspicion among some party members that power in the SNP is too concentrated with a small coterie close to Ms Sturgeon and her husband, who has held the role of Chief Executive since 2001.
However, internal criticism of the First Minister has stepped up noticeably amid a row over how both the party, and the Scottish Government, dealt with the allegations against Mr Salmond.
Mr MacAskill had previously suggested in the wake of the 2017 General Election that Mr Murrell should recruit a deputy and implement a succession plan for his eventual departure.
The former Justice Secretary also used a Scotsman column earlier this month to criticise the inner circle in the party surrounding the party leadership.
Speaking to Border TV earlier, the party grandee said that the time had come for Ms Sturgeon’s husband to ‘move on’.
He said: “The actions of the Scottish Government, as the courts decided, were cack-handed and wrong.”
Mr Macaskill also said that he didn’t believe a husband-and-wife team at the top of an organisation would be allowed in the public sector, business, or the media.
Asked whether he felt Nicola Sturgeon or Peter Murrell should step down over the case, Mr MacAskill said: “Both are outstanding individuals, both I know, and both have contributed greatly.”
“You’re not going to lose the First Minister, so I think the time has now come for the Chief Executive to move on.
“He’s got a lot of talent, his talents will be wanted elsewhere, but it’s time for there to be a separation so that the party Chief Executive is there for all the party, and there isn’t any suggestion that he’s there for his spouse.”