A group representing more than 350 secondary schools says that only a handful of its members will conduct the tests in English, mathematics and science and report the results to the Government, and that the take-up could be as low as 5 per cent.
Leaders of the main headteachers' organisations have backed the view of the National Co-ordinating Committee on Learning and Assessment (NCCLA), of which 350 secondary schools are members, that as few as 30,000 of the 600,000 14-year-olds in England and Wales may take the tests.
Last year, the three main teachers' unions boycotted the tests in protest at the extra workload they created and the fact that their results could be used to compile league tables. They were cancelled in all but a few hundred of the country's 4,000 secondary schools, and in response Mr Patten agreed to slim down both the national curriculum and testing. The National Union of Teachers is the only union still boycotting, but it now looks as if other teachers are not prepared to carry out the tests this year.
Mr Patten faces a difficult decision on whether to let the matter drop or to take action against teachers who boycott the tests. Although there is no statutory duty on them to carry out the tests, they may be in breach of their contracts of employment if they refuse to do so.
Few local authorities seem likely to dock their pay for the offence, although Wandsworth, in south London, is recruiting retired teachers and inspectors to carry out the tests and individual schools in other areas may follow suit.
Although almost all schools have requested copies of the test papers, few are expected to complete marking and reporting the results. Some will be unable to do so because of the boycott by the NUT. Others have decided that they do not like the tests or that their results could be used in league tables.
David Martin, headteacher of Chenderit School in Oxfordshire and convenor of the NCCLA, said that very few of the schools in the group would carry out all the tests and report the results. He added that his school would not test children unless their parents specifically asked for them to be tested. So far, no one had requested this.
He said: 'We still feel that the tests are unreliable and invalid, and that nationally we could do a lot better than this. They are short tests which don't give you any help in the process of diagnosing children's problems.'
The cancellation of the teachers' boycott by two of the three main unions had had little effect, he said. In many schools, headteachers and staff from all unions were in agreement that the tests should not go ahead.
David Hart, general secretary of the National Association of Head Teachers, said some schools would simply cancel all the tests, while others would do them but would not forward the results to their local authorities.
Some headteachers who have spoken out on the issue have received strongly worded letters from the Department for Education reminding them of their obligations. The subject that has proved most controversial is English, with compulsory papers on Shakespeare for all but the least able pupils.
Anne Barnes, general secretary of the National Association for the Teaching of English, said some English teachers might use some of the papers, though they would not necessarily carry out the tests this week and the results would not be reported.
'I think most English teachers will boycott the tests. Whatever happens it will be a hopeless muddle,' she said.
Tests for seven- and 11- year-olds are also likely to yield a low return from schools this year, as most NUT members are in primary schools. Schools were given from February until June to complete tests for seven-year-olds, and from April until July to complete those for 11-year-olds, which are voluntary this year.