As many as 140 schools are planning to opt out of compulsory national tests for 11 year olds.
Children are under too much stress, teachers claim
Head teachers in the East Riding of Yorkshire want to replace exams with assessments carried out by teachers.
But the Department for Education and Skills said the tests were "here to stay" and any opt-out was "unlikely".
Members of the National Union of Teachers are currently voting on whether to boycott all tests for seven, 11 and 14 year olds in England.
David Mattinson, head teacher of Elloughton Primary School, submitted the application for teacher assessment, citing the stress tests caused pupils.
TESTING IN ENGLAND
Key Stage 1
End of Year 2 - age 6/7: national tests in English and maths, marked in school, school's results available locally. From 2004, pilot scheme putting greater emphasis on teachers' assessments
Key Stage 2
End of Year 6 - age 10/11: national tests in English, maths and science, marked externally, school's results published nationally
Key Stage 3
End of Year 9 - age 13/14: national tests in English, maths and science (and ICT from 2004), marked externally, school's results published nationally
He added: "The current system is a one-off snapshot of their performance on a single morning.
"Everywhere else in education we have coursework."
The head teachers propose phasing out literacy tests for 11 year olds next year, followed by maths in 2005 and science in 2006.
Mr Mattinson said: "At the moment, children are getting a dry curriculum, where their seven years of primary school work are tested in just three or four hours."
A spokesman for East Riding of Yorkshire Council said: "As an authority we support in principle this initiative to explore alternatives to the current statutory testing framework.
"We worked with a group of head teachers to develop the proposal which was submitted to the DfES Innovations Unit.
"If this is successful we will support schools by offering training and moderation of teacher assessment."
Last week, Education Secretary Charles Clarke said: "Anyone saying we can somehow abandon [tests] is living in Alice in Wonderland.
"I don't agree with using children as pawns to pursue a political agenda, as some are doing."
The East Riding heads are attempting to use the "power to innovate", introduced in the Education Act of 2002, to avoid next May's tests.
But a DfES spokesman said: "Proposals which seek to raise standards by strengthening the accountability framework or by making the framework less bureaucratic would be welcomed, but the tests are here to stay."