Instead, there will be a new teacher assessment of four and five-year-olds when they start infant school.
The move will help to “reduce the burden” of assessment on teachers and pupils, the Department for Education (DfE) said.
One union leader said the possibility of ending Key Stage 1 testing was “good news” and would give schools more time to focus on teaching, rather than “high-stakes assessment”. However another said it might not happen until the 2020s.
Education Secretary Justine Greening said: “The Government has reformed the primary school system to make sure children can master the basics of literacy and numeracy so they get the knowledge and skills they need to succeed in later life.
“Now we want to build on that by developing a stable assessment system that helps children learn, while freeing up teachers to do what they do best - supporting children to fulfil their potential.”
Under the plans, the tests in reading, writing, maths and science, which are taken by more than half a million youngsters each year, will no longer be statutory. The proposed new baseline assessment will take place during a child’s reception year, but pupils should not know that they are being tested, the DfE said.
The results will be used as a marker of children’s abilities at the start of their schooling and be used to measure their progress aged 11, at the end of primary school.
It means that schools will be held to account for the progress that children make throughout their primary school career.
Russell Hobby, general secretary of the National Association of Head Teachers, weclomed the consultation, adding: “This creates the time and space in a pupil’s primary years for teachers to focus on teaching rather than on high-stakes assessment.
“Overall, minimising the number of high-stakes tests is the right way to go.
“This will help every school to deliver a rich educational experience for all children.”
Teaching unions have long called for an overhaul of primary school tests saying seven-year-olds should not be put under so much pressure so young. Kevin Courtney, general secretary of the National Union of Teachers, said Ms Greening “has been listening - but only partially.”
And he questioned whether testing a five-year-old would provide the forecast the DfE expected of its performance aged 11. He said: “In fact there is a wealth of evidence that points the other way.”
SATs tests for seven-year-olds will go ahead this year, with improvements, including changing the type and difficulty of questionss at the beginning so as not to discourage children by tough questions early on.
Shadow education secretary Angela Rayner said: “This consultation seems to be setting the stage for yet another embarrassing climbdown in just the latest sign of a government in chaos over education policy.
“Test papers have been leaked online, assessments have been scrapped at the last minute, and last year only half of all children met the expected standards in their SATs. It is no wonder they are now having to consult on scrapping some of the tests.”