MSP Ash Regan called for “decisive action” after Ms Sturgeon was arrested by police and released without charge on Sunday as part of an investigation into SNP finances.
The former party leader said she was “innocent of any wrongdoing”.
An SNP spokesman said the party was co-operating fully with the investigation.
Police Scotland is investigating what happened to £660,000 of donations given to the SNP by independence activists for use in a future independence referendum campaign.
Ms Regan, who quit as community safety minister over the gender recognition reform bill and finished third in the SNP leadership contest, told BBC Radio’s Good Morning Scotland that Ms Sturgeon should resign her party membership.
“There is precedence in the party for people involved in issues of this type to resign from the party voluntarily and suspend their membership until it is cleared up,” she said.
“If she [Ms Sturgeon] did that, I think it would reaffirm her commitment to the principles of the party.”
Asked if the former first minister should be suspended if she does not quit the party, Ms Regan added: “I think Humza should consider it under those circumstances, yes, if she doesn’t resign.
“We do have a code of conduct in the SNP, which says members should refrain from conduct likely to cause damage or hinder the party’s aims.
“I think Nicola will no doubt be considering whether to resign from the party at the moment.”
SNP MSP Michelle Thomson has also called for Ms Sturgeon to resign the party whip.
“This is not because she doesn’t deserve to be treated as innocent until proven guilty – she does, but because her values should be consistent,” a statement read.
Thomson, a former SNP MP, resigned the party whip in 2015 following an investigation into alleged mortgage fraud.
She later said she was given no choice but to resign by the party and criticised her treatment by its leadership.
In August 2017 the Crown Office confirmed there was insufficient evidence to launch criminal proceedings.
Meanwhile SNP MP Angus MacNeil said “this soap-opera has gone far enough”.
“Nicola Sturgeon suspended others from the SNP for an awful lot less,” Mr MacNeil tweeted.
Scottish Conservatives chairman Craig Hoy said Mr Yousaf must “now show some leadership and suspend his predecessor from the SNP”.
‘Secrecy and cover-up’
Scottish Labour deputy leader Jackie Baillie also said Ms Sturgeon should be suspended.
She told BBC Radio 4’s Today programme: “The question in my mind is, given all this chaos, given the kind of secrecy and cover-up that has been the hallmark of how the SNP operate, is whether Humza Yousaf, the current first minister, is indeed strong enough to suspend her and protect the party.”
Ms Sturgeon was taken into custody and questioned by detectives at a police station after she attended voluntarily shortly after 10:00 on Sunday.
She was released from custody at about 17:25 on the same day.
Police said a report would be sent to the Crown Office and Procurator Fiscal Service.
“I know that this ongoing investigation is difficult for people, and I am grateful that so many continue to show faith in me and appreciate that I would never do anything to harm either the SNP or the country.”
She went on: “Innocence is not just a presumption I am entitled to in law. I know beyond doubt that I am in fact innocent of any wrongdoing.”
It follows the arrest of her husband, former SNP chief executive Peter Murrell, on 5 April by officers who searched the couple’s home in Glasgow as part of their Operation Branchform probe.
The SNP’s headquarters in Edinburgh were searched on the same day and a luxury motorhome valued at about £110,000 was also seized by police from outside the home of Mr Murrell’s mother in Dunfermline.
Almost two weeks later, the party’s treasurer Colin Beattie was arrested.
Both men were released pending further investigations, with Mr Beattie resigning as treasurer a short time later. Mr Murrell quit as chief executive in March after after taking responsibility for misleading the media about party membership numbers.
Mr Yousaf said in April that Mr Murrell would not be suspended from the SNP because he is “innocent until proven guilty”. But the SNP leader confirmed the party would not be paying Mr Murrell’s legal fees.
Several SNP politicians who faced police investigations under Ms Sturgeon’s reign were suspended or had the whip removed – which automatically leads to suspension.
Her predecessor Alex Salmond quit the party in 2018 after it emerged Scottish government staff members had complained about his behaviour when he was first minister. Police Scotland said it was assessing the case at the time. Mr Salmond was later cleared at trial of sexually assaulting nine women.
SNP MP Patrick Grady was suspended from the party for making a sexual advance to a teenage member of staff. He had the whip restored in December following a six-month suspension.
Former finance minister Derek Mackay was suspended from the SNP after admitting he “behaved foolishly” by messaging a 16-year-old boy on social media. A police investigation concluded there was “nothing to suggest that an offence has been committed”.
Margaret Ferrier lost the SNP whip after speaking in the Westminster Parliament while awaiting the results of a Covid test in September 2020, before travelling by train back to Glasgow instead of isolating.
She was later ordered to complete a 270-hour community payback order by a court after admitting culpably and recklessly exposing the public “to the risk of infection, illness and death”.
Ms Sturgeon was among those who consistently called for Ms Ferrier to resign as an MP. She has now been suspended from the Commons, a move which is expected to trigger a by-election.
Another former SNP MP, Natalie McGarry, withdrew from the party whip in 2015 amid a police investigation into her finances.
She was sentenced to two years in jail for embezzling £25,000 from the SNP and a pro-independence group. The sentence was later cut to 20 months.