Allison Hunter, head of the opposition on Glasgow City Council, told the Sunday Herald her future role was "up in the air" as she may stand aside as SNP group leader.
Labour claimed the situation was "chaotic" and said voters had a right to know if Hunter wanted to run the city and its £2.5 billion budget.
Glasgow is the SNP's top target on May 3, and victory would be regarded as a significant boost to its hopes in the independence referendum.
But Hunter, 70, a former election agent to SNP deputy leader Nicola Sturgeon, has emerged as an Achilles heel, blundering repeatedly.
She admitted in an interview that she hadn't thought of any policies to implement if the SNP replaced Labour at the City Chambers.
She also said an SNP win would be a "stepping stone to independence", prompting Labour to claim her priority was the constitution, not Glasgow.
In a recent hustings, Hunter admitted the SNP Government's policy of cutting and changing bus subsidy "might be anti-city" and hurt Glasgow.
Scottish Labour leader Johann Lamont now refers regularly to Hunter on TV and radio as an example of poor quality SNP leadership.
Glasgow Labour will next week issue new leaflets featuring Hunter's quotes, saying she is no match for council leader Gordon Matheson.
The SNP group in Glasgow is due to hold its annual general meeting (AGM) between polling day and May 17, when the newly-elected council meets for the first time.
The AGM elects a group leader each year – normally a formality if the incumbent wants to stay on – and if the SNP win power, the person elected would also be the new council leader.
But in an interview with the Sunday Herald, Hunter, who only became SNP leader last June after her predecessor was elected to Holyrood, repeatedly refused to confirm whether she would stand for re- election at the AGM. "I may or may not put my name forward," she said. "I may or may not be elected if I choose to put my name forward. I mean, it's up in the air. I can't say just now because I'm so engrossed in the campaign."
The comments will increase the pressure on Hunter to quit. There are mutterings within the SNP that Hunter should go, with group business manager Graeme Hendry widely touted as a replacement.
One veteran SNP member in Glasgow said of Hunter: "She has just not done well enough during this election campaign to get the leadership. She must know it too – she's not stupid."
Another senior SNP source said of Hunter's future: "It's something that will have to be looked at. She's leading the group at the moment, but we will see what happens at the AGM."
Asked about those in the party wanting her to go, Hunter said: "The campaign is not about me. The campaign is about the team, and the team we have built up is a very good team and they're doing a very good job. I'm very happy with that."
Gordon Matheson said: "I am 100% committed to this job, to leading our city and changing Glasgow for the better. Being Leader of Glasgow City Council is a serious job that comes with a serious budget.
"Those who aren't interested in it shouldn't apply."