Lancashire County Council is advising schools not to open to more pupils from next Monday, citing fears of contributing towards a second coronavirus peak locally.
The local authority said "the safety and wellbeing of children, their families, and school staff outweigh the benefits of opening schools on June 1".
It comes ahead of the Government's decision on whether to reopen primary schools in England to Reception, Year 1 and Year 6 from next week.
"The test and trace programme is not at a state of readiness to respond to Covid-19 community setting outbreaks in a timely manner, which poses a risk to school environments.
On Sunday, Prime Minister Boris Johnson said the Government was pushing ahead with the plans, but the scientific evidence will be reviewed before a final decision is taken on Thursday.
Ministers have come under pressure from education unions and councils who have urged the Government to reconsider its plans to open schools more widely.