The SNP is privately predicting it could double the number of councils it runs in next month's polls, ousting Labour as the traditional party of Scottish local government.

Party strategists believe the SNP could leap from holding power in 13 authorities to up to 26 on an exceptional night for Alex Salmond. The SNP has set up a six-member "flying squad" of negotiators to help its councillors strike coalition deals during the crucial weekend following the results. "Time will be of the essence," admitted one of those planning for the aftermath of May 3.

Besides a long-term decline in Labour voting, the SNP is hoping to capitalise on a collapse in support for the LibDems as a result of their coalition deal with the Tories.

At the last council election in 2007, Labour and the SNP were virtually neck and neck. Labour won the biggest share of the vote, with 28.1% of first preferences to the SNP's 27.9%, although the SNP won 363 seats to Labour's 348.

This year the SNP are the only party fielding more candidates than in 2007, with hopefuls standing for half the 1222 seats.

The LibDems and SNP currently help run 13 of Scotland's 32 councils. Labour and the Tories are each in power in 11, and independents are part of the administration in a dozen. If the LibDem vote collapses as it did at the Holyrood election – when the party lost 11 of its 16 MSPs – the SNP could be the main beneficiaries.

However a lack of LibDem councillors could also rob the SNP of possible coalition partners, and see more anti-SNP administrations patched together by Labour, the Tories and Independents.

Among the councils the SNP believes it can win outright are Aberdeen, Angus, Dundee, Glasgow, Moray and Perth & Kinross.

SNP number-crunchers also believe they will end up running Aberdeenshire, Argyll & Bute, Falkirk, South Lanarkshire and Midlothian, either in coalition or as minority administrations.

To secure the maximum showing, a team of six party activists will be on call to aid coalition talks. For instance, in Labour-held Glasgow, where 40 councillors are needed for a majority, the SNP believes it could squeak home with 40 or 41.

Failing that, the SNP and Labour could have around 38 each, meaning a race to assemble a coalition with the handful of Green, LibDem and Tory councillors holding the balance of power. In such a situation, the central negotiating team would step in to help seal a quick deal.

One SNP strategist said: "Labour are in for a surprise again. Their vote seems to be holding up, but what they don't realise is that everybody outside the Labour vote hates them. Like last year there's going to be a tactical vote against them.

"We could have a leading role in 20 councils, a minor role in six, and no role in six."

Ross Martin, policy director at the independent think-tank the Centre for Scottish Public Policy, predicted the SNP would be in power in all seven Scottish cities: Aberdeen, Dundee, Edinburgh, Glasgow, Inverness (Highland Council), Perth (Perth & Kinross) and Stirling.

The implications for Labour were huge, he said. "It would signal the need for a fundamental review of what Labour stands for and seeks to represent in Scotland." Martin also predicted a rise in independent and local-issue candidates, as voters turn away from mainstream parties, a factor which could yet help the SNP.

"The message is anti-status quo. So if independent councillors are looking to properly reflect the view that got them elected they will be supporting a change of administration."

Renfrewshire North and West MSP Derek Mackay, the SNP's campaign manager, said: "We are already the party of national government and we want to be the party of local government."

But Labour MSP James Kelly said: "It is desperate arrogance and the Nationalists are guilty of taking voters for granted. It is also deeply disrespectful to proclaim victory before a single vote has been cast."