Boris Johnson admitted tonight that 1 June primary reopenings "may not be possible for all schools" but insisted his timetable remains "on track".
The prime minister said a final decision would not be taken until Thursday, which will give schools less than four days' notice.
His acknowledgement that some schools may not be able open from next Monday to Reception, Year 1 and Year 6 pupils, as he had hoped, follows a series of blows to the plan.
It has been opposed by teaching unions, scores of local authorities have failed to insist on it, and the government's own scientific advisers have warned that it should not proceed without a full track and trace system in place.
Coronavirus: Years 10 and 12 to return in three weeks
School openings: Teachers demand more evidence
Mr Johnson also used his appearance at the daily Downing Street briefing to reveal that the timetable for bringing Years 10 and 12 back into school has been put back two weeks.
Explaining why the phased reopening of schools should go ahead, Mr Johnson said: "The education of our children is crucial for their welfare, their health, for their long-term future and for social justice.
"And so in line with the approach being taken in many other countries, we want to start getting our children back into the classroom in a way that is as manageable and as safe as possible.
"We said we would begin with early years' settings and Reception, Year 1 and Year 6 in primary schools.
"Today, I can announce it is our intention to go ahead with that, as planned, on 1 June, a week on Monday.
Coronavirus: Safety fears about reopening schools
"We then intend from 15 June for secondary schools to provide some contact for Year 10 and Year 12 students to help them to prepare for exams next year, with up to a quarter of these students in at any point."
He said the government was being "deliberately cautious" and that a final decision would be taken next Thursday as part of the formal review into the lockdown measures.
The prime minister added: "I acknowledge that the 1 June opening may not be possible for all schools but the government will continue to support and work with the sector so that any schools experiencing difficulties are able to open more widely as soon as possible."
Kevin Courtney, joint general secretary of the NEU teaching union, said that it was still opposed to the 1 June reopenings.
"We once again call on the government to engage meaningfully with the education unions on these matters," he said.
"We stand ready to talk to the government about how our five tests can be met and then how we can then proceed to a safe wider reopening of schools."
The Association of School and College Leaders said it welcomed the government's recognition that not all schools would be able to open in a week's time.
But Geoff Barton, ASCL's general secretary, said: "[The government] has not communicated the rationale for its chosen approach well, and it left primary schools with little time to plan and implement safety protocols.
"It is also worrying that the government's crucial test, trace and isolate system is not yet in place and is unproven."
He added: "It is doubtful that any part of the education sector has ever been asked to do so much in so little time."
Teachers' union the NASUWT said the decision to push ahead with the 1 June opening was "at odds" with scientific evidence on transmission.
General secretary Patrick Roach said: "The government has to recognise that it has not won the trust and confidence of the teaching profession.
"Notwithstanding the government's assertions, the bottom line is that no teacher or child should be expected to go into schools until it can be demonstrated that it is safe for them to do so."
Paul Whiteman, general secretary of school leaders' union the NAHT, said: "That the prime minister acknowledges that flexibility will not just be possible but will be necessary is to be welcomed.
"We will take the prime minister at his word that schools will be allowed to react to their own local situations and will not be forced into opening or penalised if proceeding with appropriate caution."
Education secretary Gavin Williamson said: “This has been a difficult time for the entire country. Education and childcare staff have stepped up to the challenge, making sure children and young people have continued to be supported throughout the past 10 weeks.
“Our priority is the education and welfare of all children and young people across the country. That is why we want to start a phased wider opening of nurseries, schools and colleges informed by the best possible scientific and medical advice.
“We will continue to work with the sector to support them to prepare for wider opening and ensure all children and young people can continue to receive the best care, education and training possible.”