More than 4,700 schools are taking part in a pilot project this year which allows them to testseven-year-olds when they want.
Because these schools report back only teacher assessments, not test results, children should not be disadvantaged by taking the reading and maths tests early.
Montgomery primary in Birmingham started doing the tests in January using last year's papers because some of the 90 seven-year-olds were ready to do them.
Sue Rose, deputy head and key stage 1 co-ordinator, said the pilot scheme had made organising the tests much easier. "It has made a huge difference," she said. "Rather than wait until May and have a mad rush in that month, we do the tests as we go along. There will be about 20 children who have not completed them by May and they will then do this year's tests."
The 630-pupil primary has 95 per cent of children with English as an additional language and 35 per cent on free school meals. Other schools have decided not to take up the option of earlier tests.
Steven Nairn, county adviser (assessment) for Kent, advised all 411 schools taking part in the pilot to test in May, because he did not want test results to influence teacher assessment.
He said: "We advised schools to finalise their teacher assessments towards the end of the spring term and then use the test to inform that teacher assessment. We were fearful that schools may have been tempted to use last year's test to arrive at a teacher assessment mark and then seek to confirm that by using this year's test."
Rachel Denton, KS1 co-ordinator at Nafferton primary, in Driffield, Yorkshire, said one reason she was administering tests in May was because the children were more able to cope by then.
She said: "The pilot is a step in the right direction, but if the teacher assessment is the final mark why do we have to do tests at all? Why put children through it?"
Jackie Bawden, head of testing at the Qualifications and Curriculum Authority, said: "Some schools have used the tests early for diagnostic purposes, to identify strengths and weaknesses which the teacher can build on."
David Hart, general secretary of the National Association of Head Teachers, said: "If the tests are taken in January, I see no reason why teachers cannot then rely on teacher assessment in the summer in terms of the outcome that goes to parents."