National curriculum tests taken by seven-year-olds in England are set to be axed under Government proposals.
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".
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 - known as SATs - 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 at some point 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 the progress youngsters have made by age 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 (NAHT), said: "The possibility of ending Key Stage 1 SATs is good news.
"This creates the time and space in a pupil's primary years for teachers to focus on teaching rather than on high-stakes assessment.
"It will properly reward early intervention and it will reduce workload.
"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 testing, arguing that children are assessed too often, and that testing puts pressure on youngsters at an early age.
Kevin Courtney, general secretary of the National Union of Teachers, said: "Ms Greening has been listening - but only partially.
"The consultation floats the idea that statutory assessment at KS1 will be set aside, but not until the early 2020s.
"This would be a welcome concession to the thousands of teachers who have protested against the effects of a test-driven curriculum on six and seven-year-olds.
"But the relief that is offered at one stage of education is accompanied by changes for the worse for younger age groups.
"In a triumph of hope over experience, the DfE wants to reintroduce baseline testing to the early years, despite its failure in 2015/16.
"The DfE wants to believe that the test results of a five-year-old can reasonably predict their performance at 11, so that the school system can be held to account if children do not make the 'expected' progress.
"In fact there is a wealth of evidence that points the other way."
The Government is consulting on the proposals, which also includes making improvements to the early years foundation stage - which records young children's progress up to age five.
SATs tests for seven-year-olds will go ahead this year, with some improvements, including changes to the type and difficulty of questions at the start of the tests, to ensure children are not discouraged by tough questions early on.