Up to 140 primary schools are planning to exploit government legislation to scrap tests for 11-year-olds.
In the first move of its kind schools in Yorkshire have applied to use the "power to innovate", introduced by the Government in the Education Act 2002, to avoid national tests.
Ministers are considering the application from primaries in the East Riding to phase out the tests and replace them with teacher assessments.
The application, which has the backing of the local authority, comes at a sensitive time when the Government is facing a national boycott of the tests by the National Union of Teachers.
Ministers initially said they would not approve any applications to "innovate" which diluted the testing and accountability regime. An application made by Birmingham city council in July - to scrap tests and league tables for its primary schools - was rejected out of hand.
But dismissal of the latest application would run the risk of hardening opposition among teachers to the tests and increasing the likelihood of a boycott.
David Mattinson, head of Elloughton primary, who submitted the application, said that he had decided to take action after this year's tests had distressed pupils.
"We had a pretty bad experience but we have attempted to seek a positive way forward," he said.
If his application is accepted, schools that wish to take part will next year replace the writing test with teacher assessment based on a "rolling programme" of work. In subsequent years a similar approach will replace science and then maths tests.
Mr Mattinson submitted the formal application to the Department for Education Skills at the beginning of this term, but despite what he describes as an "optimistic" initial response he has not yet received a decision.
Instead the process is bogged down in a wrangle over whether the application can cover all primaries in the county without their signatures.
"It is very frustrating. I cannot realistically get signatures from 140 schools. I do not have the time," Mr Mattinson said.
A spokeswoman for East Riding said the council expected it to apply to all schools. "We worked with a group of headteachers to develop the proposal which was submitted to the DfES innovations unit.
"If this is successful then we will support schools by offering training and moderation of teacher assessment," she said.
The DfES warned that it was unlikely to allow any school to opt out of tests.
"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," a spokeswoman said.