Last week's elections brought more bad news for Scottish Labour.
With exactly one month to go until the general election, Ms Sturgeon said that without a strong SNP voice for Scotland at Westminster the Tories will think they have a free hand to negotiate any Brexit deal - or crash out of Europe without a deal - in the months and years to come.
The party has held the city since 1980, but now the SNP is hoping to win a majority.
A total of 262 councillors were elected on a Labour platform, although one of them had been expelled from the party after the ballot papers had been printed.
Despite topping the poll, the party lost overall control of Dundee and Angus councils, which it had before the election.
Last week the UK Prime Minister accused the European Union of interfering in the upcoming UK General Elections on 8 June, causing more tensions with the 27 member block and drawing criticism for provoking what some analysts fear could be a "no deal Brexit" situation.
Labour MSP Anas Sarwar said losing seats is "disappointing" for the party.
Ms Davidson said the council election results had been "very encouraging" but that her party was taking "nothing for granted" ahead of the poll on 8 June.
Labour lost all but one of its Scottish MPs in the 2015 general election, but Mr Kelly said: "Labour was written off before the council elections, but we defied the odds to win three councils, finished joint top in another, and pushed the SNP incredibly close across Scotland".More news: President Trump signs controversial 'Religious Liberty' executive order
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The group took seven of the 18 available seats, with Labour only managing four, the SNP five seats and the remaining two going to independent candidates.
The comment fits with a national Conservative decision to carefully downplay their success, although it appears many inside the party are expectant of a significant win north of the border.
Ms Sturgeon started the first full week of campaigning at Inveralmond Brewery in Perth with local SNP MP Pete Wishart, where she practised pulling pints.
Once more than half of the council's 85 seats are declared, expected to be in the next round of results, it will become clear whether the SNP can gain majority control or whether a minority administration or coalition is on the cards.
"An SNP vote is the only way to guarantee more affordable housing, a transformation in early learning and childcare, investing in our schools, more money and power for local communities and increased support for small businesses".
The Tory leader will call for a "Herculean" effort from her party as she presents herself as the opposition to SNP leader Nicola Sturgeon, saying many are "looking for somebody to stand up" to her.
She said: "We will speak up for the millions of Scots who have had enough of the uncertainty and division of the last few years".
This is the third local government election in a row that the party has made losses.
The SNP will also be keeping a close eye on East Dunbartonshire.