Tests for 11-year-olds will be ignored by thousands of schools next Monday.
Even using the most conservative estimates from England's local authorities, almost 4,000 of the 17,000 schools due to take the SATs tests look certain to cancel them.
A survey of 39 local authorities found that just over 1,000 schools had already told their employers the tests – due to be sat by 600,000 11-year-olds in maths and English – would not go ahead. If this is reflected nationwide, 3,900 schools would be hit by the dispute.
Some authorities insisted that not a single school in their area had indicated they would be boycotting the tests, while others said that up to 60 per cent were taking action. Teachers' leaders said they believed support for the action would be higher than expected.
Christine Blower, general secretary of the National Union of Teachers (NUT), one of the two unions which voted for a boycott, said up to 50 per cent of primary schools had pledged support for the action and that the number was growing. "At that level of participation in the action, it would be impossible to draw up league tables," she said.
Mick Brookes, general secretary of the National Association of Head Teachers (NAHT), the other union taking part, originally estimated that between 6,000 and 7,000 schools would support the boycott. However, he said that more were planning to join in following guidance from the Schools Secretary, Ed Balls, that school governing bodies should consider sending the headteacher home in the event of a boycott and persuading another "competent" adult to run the tests.
Of the 39 authorities which have given details of school responses, six said they had not been told of any school taking part in the boycott. These include Bracknell Forest, Bournemouth, Kingston-upon-Thames, Slough and the Isle of Wight.
A spokeswoman for the 32 London local authorities said that of the six that had given details, five would be affected. Only Islington said it could report no effect. In Tower Hamlets, the action was said to strongly supported.
In the 33 authorities saying there would be some effect, a total of just over 1,000 schools said they would stage a boycott. In Wakefield and Barnsley, Steve Iredale, local executive member for the NAHT and a primary school head, said he was anticipating 60 per cent support for the action. More than 40 authorities said they could not yet estimate the effects of the dispute.
Mr Brookes has offered what he has called an olive branch to the Government: a moratorium on testing until a new method of assessing pupils could be agreed. The suggestion was rejected by ministers before the election.
The unions claim the tests, which are used to produce primary school league tables, have led to a narrowing of the curriculum, with schools teaching to the tests to ensure a good showing in the tables. The tests are due to start on Monday and last for four days.
Both the Conservatives and Labour are committed to going ahead with the tests. Labour is prepared to replace them with teacher assessment if the results of internal assessments are as robust as external marking.
The Conservatives have promised a review of the tests, but insist they are here to stay in some form or another. The Liberal Democrats favour a move towards a more assessment-based system, with education spokesman David Laws also promising a reform of school league tables.