Party leaders have been trying to rouse voters, especially for the SNP-Labour fight for control of Glasgow

Scottish political leaders were out in force throughout polling day in a final effort to rouse apathetic voters amid predictions of a low turnout.

Despite bright sun and warm temperatures across central Scotland, senior figures in all the mainstream parties fear the turnout could hit a record low for recent local council elections, slumping to less than 40% of Scotland's 4 million eligible voters.

The most intense battle for votes was in Glasgow, where the influential Scottish National party MSP Humza Yousaf predicted his party could win a shock victory over Labour after seeing substantial support for nationalists at polling stations.

With the SNP widely expected to make further gains across Scotland, taking control of smaller cities and councils, Glasgow is the greatest prize for first minister Alex Salmond. He and party executives have been carefully playing down their early hopes of winning Glasgow, but Yousaf said their supporters were out in strength on polling day.

He told the Guardian the party was poised to wrest control of Scotland's largest city after decades of Labour domination, allowing the SNP to form a coalition or run a minority administration if it failed to build a large enough cross-party alliance.

"We're getting a good reaction. My prediction is we will be two to three seats ahead of Labour. A majority is going to be tough but I think we will get the mandate we need to form an administration," he said.

His predictions were contested by the Scottish Labour party, which is defending a 20-seat majority over the SNP in Glasgow and has been specially targeting 10 areas where the party hopes to win all or most seats in the multi-member wards.

Anas Sarwar MP, the deputy Scottish Labour leader, said he was unwilling to make firm predictions but said: "I have been in five of the key wards today and the mood has been very positive on the doors. The message coming across is about putting the city of Glasgow first, rather than worrying about an SNP independence referendum."

There was caution from Gordon Matheson, Glasgow's Labour leader and current council leader. While predicting victory in an interview with the Guardian last week, Matheson would not call the result while he visited Labour voters on Thursday.

"I'm very positive," he said. "We're at a stage where we're benefiting from all the months of effort identifying our vote."

Ruth Davidson, the Scottish Tory leader, who was out seeing Tory voters in southern Glasgow, said the race in the city was so tight it was possible that one seat could divide Labour and the SNP. Told about Yousaf's prediction, Davidson said: "I think it will be closer than that. I still think Glasgow is too close to call between the big two."

The parties are competing for 1,223 seats in 32 Scottish councils in Thursday's election. The SNP is currently the largest party, defending 368 seats and control of 13 councils through coalitions or minority administrations. After standing 612 candidates against 497 for Labour, Salmond's target is to increase that number of seats and the SNP's share of the vote by outstripping Labour.

It is widely expected that the Lib Dem vote will slump across much of Scotland: the party has only 247 candidates; the Tories have 362, Scottish Greens have 86 and691 independents and minority party candidates are standing including the UK Independence party and Solidarity.

Votes in Scotland are being counted on Friday, with the major cities expected to declare final results during Friday afternoon and early evening.