A spokesperson for the Department for Education said reading "holds the key to the rest of the curriculum, with pupils who struggle to read so often struggling across the board".
They said that was particularly true of "those from disadvantaged backgrounds".
The plans are expected to form part of an imminent schools white paper, which the spokesperson added would "set out an ambitious vision to make sure every young person, wherever they grow up, has the opportunity to succeed".
The proposed changes would mean most pupils in England sitting national tests in seven years of their school education:
Results from the tests are expected to be shared with the government and Ofsted, and not published for individual schools.
However, Daniel Kebede, General Secretary of the National Education Union, said there was "nothing to stop future governments publishing them school-by-school".
He said that would lead to "punitive labelling of schools", more work and "increased stress" for staff and students - as well as a "narrowing curriculum" geared towards tests rather than being "broad and balanced".
"It is beyond belief that this government's response to students disengaging in secondary isn't to consider the impacts on curriculum caused by the tests that already exist in primary, but rather is to suggest an additional test in Year 8," he said.
"The government needs to understand that young people being churned through test after test after test does not automatically equate to high standards."
The outcome of a government-commissioned curriculum and assessment review is also due to be published soon.
Pepe Di'Iasio, general secretary of the Association of School and College Leaders, said "many schools already assess Year 8 pupils to check for learning gaps" and a standard test, if "done well", could "help to support a consistent approach".
"However, it is important that this does not end up becoming another accountability measure – either through the publication of results or as part of Ofsted inspections," he said
He added that school leaders had "been bitten too many times".
Sarah Hannafin, head of policy for school leaders' union NAHT, said the reading test would be "unnecessary, distracting and not a good use of money when funding is tight for schools".
"Taking a test does not improve standards; good teaching does," she said.
"This money would be much better spent on resources to support teaching and learning strategies to improve reading."
This year, 75% of Year 6 students achieved the expected standard of reading in their SATs exams.
Nearly a quarter of maths and English GCSEs were taken by people aged 17 and older this year. They made up 23.4% of maths and English grades – compared to 20.9% last year.
Some would be mature students sitting exams for the first time, but most would have been young people resitting papers.