Parents in England who do not send their children back to school in September will face fines, says the Education Secretary Gavin Williamson.
"Unless there's a good reason for absence… we'd be imposing fines on families," he said.
But head teachers said fining parents was not the "right approach" at first.
"There will be many frightened and anxious parents out there," said Geoff Barton, leader of the ASCL head teachers' union.
Head teachers, who decide whether absences are authorised, are more likely to want to build up parents' trust in a safe return, said Mr Barton.
Mr Williamson, speaking on LBC, said penalty fines for non-attendance would be part of school being compulsory again next term, unless there were "good reasons" such as a local spike in infections.
"We do have to get back into compulsory education and obviously fines sit alongside as part of that," said England's education secretary.
This threat of fines was criticised by the Young Minds mental health charity, which warned that many young people could feel "extreme anxiety" about going back to school, such as if they were worried about shielding a relative with health problems.
Instead of issuing fines, the charity suggested a "transition period" with more flexibility at the beginning of term.
How did parents respond?
"Totally in favour of children returning to school now and not a fan of parents taking their kids out of school," said Chris, responding to BBC over this story. "But this is a joke."
He says if children can be out of school for six months - "how does missing a few days of school suddenly become a huge deal?"
"Sorry but that is so rude!" said Shelley. She said her children have been out of school for months - "and suddenly I am being threatened with fines?"
"Are you joking? You can't flip from expecting us to take the reins to threatening us!" said Shelley.
"Not a problem. Now I can fine the school for not providing any education since the lockdown," said another parent, with a daughter who had only had 30 minutes face-to-face time at her secondary school.
"So despite being a parent who has struggled to maintain working from home alongside full time home schooling for the past four months and having to create my own curriculum, the loudest voice we hear from the education secretary is threats of fines," said Kevin in Exeter.
Mick in the East Midlands said: "I think that fines are a too heavy handed approach. There should be a period of time for the school, parents and students to settle in."
"If the epidemic spikes in September, after the government easiness in the summer, then there is no way to send my child to school," said Mo in West London.
Chris in London said children will soon be able to "go shopping on Oxford Street, or to Spain on holiday, but not to school. This is a disgrace".
'It's not about 1 metre...'
Mr Williamson also indicated on Monday that the return to school in the autumn would not rely on social distancing.
"It's not about 1 metre, it's not about 2 metres," he told BBC Breakfast, saying that safety would be based on "reducing the number of transmission points" within schools.
This would mean whole classes becoming "bubbles" separated from other pupils - and he promised "comprehensive plans in terms of both testing and tracing".
That could mean local closures of schools or sending home individual year groups in response to Covid-19 infections.
During the return to school for some classes during the lockdown, attendance has been voluntary and fines have been suspended.
Under this voluntary arrangement, in primary school only about a third of Year 6 pupils are attending and a quarter of pupils in Year 1.
But all pupils are required to go back to school full-time in September - and the fines for unauthorised absence will also be applied.
'Period of grace'
Head teachers will decide if an absence is unauthorised - and this will be referred to local authorities, who can issue a fixed penalty notice of £60, rising to £120 if not paid within 21 days.
Head teachers rejected the idea of an immediate issuing of fines, suggesting a "period of grace" to build up trust over safety at the start of term.
"This is very much a case of building confidence that it is safe to return, rather than forcing the issue through the use of fines," said Mr Barton.
Robert Halfon, chair of the Education Select Committee, was also doubtful of imposing fines from the start of term.
He said some families would have "genuine fears" over safety, and rather than issuing fines, he proposed a compulsory attendance register to be operated by local authorities, who would regularly contact families to ask why children were not in school.
Labour leader Sir Keir Starmer backed the compulsory return to school in September.
"It absolutely should be the right thing that children are in school and everything should be done to get them into school," he said.
But Sir Keir accused the prime minister of "trying to blame everyone else" for why there were not more pupils in school already.