There were no new cases confirmed in Tasmania on Tuesday, keeping the state's total at 226.

About two-thirds of the state's total recorded cases have been in the northwest, with 12 of the state's 13 virus-related deaths in the region.

The good news came hours after Premier Peter Gutwein said he was hopeful of making a call in July after reopening the border.

"If we can continue to follow the rules, I expect that in July we will be able to set a date for when our borders could come down," he said.

Federal Tourism Minister Simon Birmingham has urged state and territories to considering opening their borders to domestic visitors if COVID-19 case numbers stay low.

The island state has moved to stage one of a plan out of restrictions on Monday and is expected to begin stage three in mid-July.

A coroner is investigating the deaths of two people in the northwest who tested positive but also had serious pre-existing health conditions.

The man and woman, both in their 60s, died on Sunday and Monday respectively at the North West Regional Hospital in Burnie.

The pair had been diagnosed with coronavirus in late March.

"Both had progressive life-threatening conditions that were likely to lead to death in the short to medium term," Public Health Director Mark Veitch said.

The North West Regional Hospital and its private counterpart were shut last month after scores of staff and patients tested positive to the virus.

Both of the people were in the NWRH at the time transmission was occurring, Dr Veitch said.

The facilities returned to full operations about a week ago after a specialist deep clean.

The deaths won't be counted as part of the official COVID-19 death toll until more information is collected.

The state government announced support of up to $2000 for people struggling to pay rent, at an overall expected cost of $1.5 million.

Australian Associated Press