Premier Peter Gutwein says he is hopeful schools in the state's North-West will reopen next week.
As part of tough restrictions in the region following an outbreak of COVID-19 at the North West Regional Hospital, Term 2 in the North-West has been delayed by at least four days.
Coronavirus: All the latest updates on COVID-19 for Tasmania
Mr Gutwein said the state's plan for schools was clear.
"Our position remains consistent with what we announced before Easter - our schools will remain open. Parents should keep their children home unless they cannot support their learning," Mr Gutwein said.
"Our schools in the North and the South are open, but as we've said we've planned for Term 2 to be different."
Mr Gutwein said the government would continue to take the advice of health authorities when considering lifting restrictions in the North-West.
He said he would be announcing an economic and social recovery plan for the state later this week.
"We are seeing a way forward through this, importantly, and the low numbers [of cases] provide evidence that we do not as yet have any widespread community transmission," Mr Gutwein said.
"As we step our way out of this we need to do so sensibly and responsibility to make sure that older age cohort we have in Tasmania is protected as well."
In other news:
There have been 214 cases of COVID-19 in Tasmania of which 135 people have now recovered.
Health Minister Sarah Courtney said there was a further possible case in Southern Tasmania.
"I'd like to be clear this person has not tested positive. They have a borderline test," Ms Courtney said.
She said the person would be retested, with the results to be known on Wednesday, and the person's close contacts were being contacted and asked to quarantine.
Public Health deputy director Scott McKeown said a precautionary approach was being taken towards this possible case.
Dr McKeown said this person had been in quarantine since they became unwell.
"This is the first time in Tasmania [we've had a borderline test] and we are investigating," Dr McKeown said.
744 people were tested for COVID-19 on Monday of which 77 were in the South, 48 in the North and 619 in the North-West.
"We'd really like to see this [increased level of testing] continue," Dr McKeown said.
"Our key public health message is at the moment - anywhere in the state, if you have the mildest of symptoms ... please get tested."
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