The SNP - which has been in power in the Scottish Parliament for a decade - has 267 councillors, putting Nicola Sturgeon's party down by one.
"The Conservative vote is on the rise, but I think at the end of the day Scotland will be the only place the Conservatives have been beaten - that is to say the objective of an election campaign is to win more votes and more seats than any other party - and we are confident the SNP will do exactly that across 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 National party leader said that although the Conservatives did well "by their standards", they did so by taking votes from Labour, not by eating into SNP support.
Councillors Stocks said: "As far I'm concerned we won the election in North Lanarkshire and it is our right to try to form the next administration".
Professor Curtice said SNP deputy leader Angus Robertson's Moray seat was under threat while the Tories were favourites to take Calum Kerr's Berwickshire, Roxburgh and Selkirk constituency.
Nicola Sturgeon has dismissed as "ludicrous" any claims that the surge in Tory support in Scotland could derail her bid to hold a second independence referendum.
VoteSNP on 8 June to stand against Tory cuts.
The Labour collapse is also expected to give COSLA the first non-Labour president in its 42-year history, though there too there are complications.
Labour leader Kezia Dugdale described the results as " obviously a disappointing election".
Speaking on the campaigning trail, Sturgeon said: "In the last few days, the Tory mask has slipped. We know from the polls in England that Theresa May is going to win this election, the question is who is going to stand up and make Scotland's voice heard?" However, they may find that the harsh reality of a Westminster election is that 25 per cent of the overall vote, as achieved in the local elections, is not rewarded with 25 per cent of the seats. Ukip lost 145 council seats - all but one.
Labour has fallen to third place with 147 councillors, down by 59.
Nationally, May's party gained more than 550 council seats and swept to shock victories in mayoralty contests in the West Midlands and Tees Valley.
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". We have heard all of Ms Sturgeon's lines before - only the SNP will act as a bulwark against the Tories, only a vote for the SNP is a vote for Scotland, and so on.
The party polled 4.1% of first preference votes, an increase on the last election.