The move will help to "reduce the burden" of assessment on teachers and pupils, the Department for Education (DfE), said.
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.
Instead, ministers are proposing to bring in a new teacher assessment for four and five-year-olds in the reception year of infant school - the year before formal schooling officially begins.
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.
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"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."
The Government is consulting on the proposals, which also includes removing the requirement on schools to submit teacher assessments of pupils, where their assessments are not used to hold schools to account, and making improvements to the early years foundation stage - which records young children's progress up to age five.
The new assessment for youngsters in their reception year will be developed with the teaching profession, the DfE said.
Schools will also still be given SATs test materials for seven-year-olds to help them benchmark pupils' abilities and inform parents of progress.
The Government will sample schools that administer the papers to ensure academic standards stay high, the Department said.
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.
"The consultation proposes replacing the Key Stage 1 SATs with a properly designed assessment in reception in order to create a baseline for a progress measure.
"NAHT support this approach as long as this is not a high-stakes assessment for pupil or school.
"It could more fairly reflect the challenges faced by different schools. And it is possible to design it to avoid predicting or tracking individual pupil performance from such a young age."