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".
Opinion polls giving May's Tories a lead of nearly 20 points over their main rivals, the Labour Party, could be wrong because United Kingdom surveys were misleading in earlier elections, she will say, according to emailed extracts of her speech.
The Scottish Liberal Democrats received a 6.8% share of the first preference votes as they won 67 seats, with the Scottish Greens winning 19 seats on 4.1% of the votes.
The Labour vote has collapsed and gone to the Tories.
Ms Sturgeon said: "The issue at the heart of this election, not just in Perth and Perthshire but across Scotland, is a very straightforward one".More news: Syria violence kills 4, wounds child despite safe zones
"Labour values must run through any deals: the defence of local services against cuts; and the proper funding of the services so many people rely on such as education and care for the elderly". Maybe we are simply discovering that voters in Scotland are just not so keen on voting for the SNP in local elections - as was the case for Westminster elections until the party's 2015 success.
Nevertheless, the SNP (slightly) overplayed its hand in this election, giving the impression it would win overall control of local authorities in Aberdeen and Glasgow.
The first minister said it showed the only party that could possibly "stand up" to the Tories were the SNP, and Ms Davidson said it demonstrated the only party that could "stand up" to the SNP were the Scottish Conservatives.
Nicola Sturgeon has dismissed claims that the surge in Tory support in Scotland could derail her plans for a second independence referendum as "ludicrous".
In a series of eyebrow-raising results, the Tories also won a seat in Cowdenbeath, where previous year the UK's last elected Communist stood down, and even in Ravenscraig.More news: Beyond 100 days, Trump faces more legislative challenges
"They made it about that, not the SNP - and we won, so in any way of cutting an election result that says the SNP was endorsed rather than the Tories".
"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 Tories want to deprive young people of opportunities, they want to hammer working families and they want to put more of a burden on households with punitive policies like the Bedroom Tax and Rape Clause. 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".
Last week's results followed a campaign by the Tories focused on opposition to a second referendum - a message they will carry through their General Election campaign in the weeks ahead.
Tony Blair's landslide victory 20 years ago all but wiped out Scottish Conservatism, with the party failing to win a Westminster seat in 1997 and never holding more than a single constituency since. Turnout will also be higher next month, although the 46.9 per cent turnout last week was higher than normal for council votes.More news: Sharapova more disappointed to lose early than to detractor