Portsmouth City Council and Hampshire County Council both say they are following Department for Education guidance.
In Portsmouth, any penalty notices or prosecutions linked to Covid-19 absences have been suspended.
But the authority has also withdrawn new cases from March 16 onwards.
A statement said: ‘This approach was applied to prosecutions for non-attendance.
Legal proceedings are taken in arrears so any outstanding prosecutions were for offences committed in the period December to the end of February.
‘Reviews took place to decide whether to proceed with those that related to the period prior to March 16, 2020. Each case was reviewed individually as it always would be, irrespective of Covid-19.’
In Hampshire no Year 11 pupils’ parents will face legal action over non-attendance.
And penalty notices issued in March – or due to be paid in March – have been withdrawn. Any payments made will be refunded.
Councillor Roz Chadd, executive member for education and skills, said: ‘In response to Covid-19, we have suspended the issuing of all penalty notices until schools are fully open and DfE guidance is available.
‘All penalty notices issued in March, or where payment fell due in March have been withdrawn and refunded if paid, and no legal action relating to the attendance of a Year 11 student will be pursued.
‘In all other cases, due process will be followed as normal, making use of available technology for meetings, in place of face-to-face contact.’
At the moment it is planned that nurseries, Year R, Year 1 and Year 6 in primary schools, and Year 10 and Year 12 in secondary schools and colleges will resume on Monday June 1.
However, both Portsmouth and Hampshire councils have said that they are awaiting further guidance before committing to this, and many teachers and unions have protested that it will not be safe.