"The fact is you can not trust the Tories - they're cutting Scotland's budget as it stands and gearing up to inflict more austerity and further hardship in the years ahead", Ms Sturgeon added.
The SNP leader will cite the Prime Minister's claims that European Union leaders are trying to interfere in the general election to warn that the Tories want a hard Brexit "whatever the cost to Scotland".
SNP MSP Derek Mackay, who has been announced as the party's campaign director for next month's general election, said the council election results would be an "excellent springboard".
The Tories have analysed first preference votes given in the council wards of Lower Deeside, Torry and Ferryhill, Kincorth, Loirston and Cove, Hazlehead, Queens Cross and Countesswells and Airyhall, Broomhill and Garthdee.
Internal party data based on how Scotland voted at last week's local government election claims that the party is poised to stage a remarkable comeback north of the border.
The Tories took 25.3 per cent last week - more than double the 12 per cent secured by the party in 2012 and up from 22 per cent in 2016.
Ms Davidson said the results showed "only the Scottish Conservatives have the strength to fight back against the SNP".
The former first minster continued: "I've never taken any election battle for granted, I've represented the north-east of Scotland for 30 years and every election the Tories tell me they are going to beat me and every time they fall short".
The SNP remains the largest party in local government with 431 councillors voted in, up slightly from its total of 425 in 2012.
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.
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But while admitting the Conservatives had done well in Scotland, Sturgeon insisted that May's party had eaten into Labour's rather than her own SNP party's support.
However, First Minister Nicola Sturgeon has said only the SNP can "stand up to the Tories".
Nicola Sturgeon has dismissed claims that the surge in Tory support in Scotland could derail her plans for a second independence referendum as "ludicrous".
"There's no question the Tory vote is on the increase and therefore we have to galvanise the SNP support", he told Press Association Scotland. "The cliché is that it's a two-horse race, but it really is, and we will be leading Scotland's fightback against the SNP".
"Taking nothing away from the tremendous result they had in 2015, they then tried to do that thing where they made out that the SNP was Scotland and Scotland was the SNP".
VoteSNP on 8 June to stand against Tory cuts.
"Scotland must speak with a clear voice in this election to stop the Tories in their tracks".
Aitken says she is exploring a deal with Glasgow's seven Green councillors, which would give the administration 46 votes, against Labour's 31 and the Conservatives' seven.
Labour also had a higher percentage of first preference votes than the SNP in Coatbridge, Chryston and Bellshill, up 1.65% percentage points, and was 0.91 percentage points higher in Lanark and Hamilton East.