Mrs Maginnis said she was "not surprised" by the massive rejection of the outline deal by the Scottish Secondary Teachers' Association, nor the warning to members this week by the Educational Institute of Scotland, the lead union, that despite progress "significant problems remain".
Mrs Maginnis said: "The EIS strategy is very clear. They want to wait until after the Scottish parliamentary elections. I would not be surprised if there is a speedy conclusion to the negotiations once people see a Labour majority."
She added: "The only way the EIS could pretend the Scottish Joint Negotiating Committee has a future is if it delivers the agenda for change."
Unions have been warned it may be August before talks resume because of difficulties in reconstituting a management negotiating team after the elections. Some council leaders outside the talks already believe their negotiators have conceded too much ground.
Only 1.76 per cent of SSTA members accepted the employers' offer in a ballot with a 65 per cent turnout. Strikes could follow.