Introducing the new baseline assessment in September may not be "practical or sensible", as it is "unlikely" schools will be operating as normal, heads have said.
The Association of School and College Leaders (ASCL) has said it hopes to reach "a sensible way forward" with the government regarding baseline, to "avoid a situation in which anybody feels the necessity to boycott the assessment".
More than 2,800 people have signed an open letter to schools minister Nick Gibb, calling for the Reception baseline assessment (RBA) to be scrapped due to disruption caused by the coronavirus outbreak.
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The 20-minute assessment, which is supposed to be carried out within six weeks of the new term, is due to be rolled out across all primary schools from September this year.
Some headteachers have said they are considering boycotting the RBA, as they are concerned it will have a negative impact on children's wellbeing and development.
Geoff Barton, ASCL general secretary, called for the plan to begin the assessments in September to be kept "under review".
He said: "It seems unlikely that schools will be back to operating as normal by then because of health and safety considerations over social distancing, and it may not be practical or sensible to undertake Reception baseline assessments in those circumstances."
"We would hope to avoid a situation in which anybody feels the necessity to boycott the assessment by reaching a sensible way forward with the government on this issue."
Chris Dyson, head of Parklands Primary School in Leeds, said he is planning to boycott the assessment in the autumn.
"These kids are going to come back to school in September and they will need nurturing and their wellbeing looked after," he said.
"This has been an unprecedented piece of history. We don't want to have these kids back in and say: 'We are testing you now.'"
Mr Dyson, who has called for more schools to join him in boycotting the assessment, added: "At this very moment of time, we probably won't be doing it. And I will be saying the reason why we are not doing it is because we are looking after children's wellbeing and their state of mind."
The RBA, which evaluates children's skills in English and maths at the start of Reception, is designed to provide a starting point for schools so they can measure pupils' progress.
But many of the children starting primary school for the first time in the autumn are likely to have been out of nursery, or other early years provision, for a number of months.
Mark Chatley, head of Palace Wood Primary School in Maidstone, Kent, said he is also considering boycotting the assessment in September if no changes are made.
He said: "Schools should be working on nurturing the child on what is normal, developing their wellbeing and resilience, and helping them feel part of a community, rather than taking them out of a classroom one-to-one with a person they have only known for a short period of time and doing questions on maths and English."
Mr Chatley added that schools will have to prioritise building up pupils' physical development at the start of term as some children will have spent less time in play areas during the shutdown.
When asked if he would boycott the assessment, he said: "Every day that passes, it becomes much more likely that we won't do it.
"I can't see the way that schools are going to be back before the May half-term break in any capacity and nurseries will be very similar.
"If it were to be done, I think school leaders – and I would include myself within this – would need to be really strong and say what we actually need to do right now is focus on what is right for our children."
Mary Bousted, joint general secretary of the National Education Union teaching union, which has signed the open letter, said: "The last thing that we need this September is a new test, especially for very young children.
"After the disruption of the pandemic, 4-year-olds will need a carefully managed period of settling into their new school. With baseline assessment, this would not happen."
A DfE spokesperson said: "We understand the challenging circumstances schools are facing and so are working closely with our delivery partners NFER to keep the progress of the Reception baseline assessment under review."