Thousands of people across the country attended Invasion Day marches and protests on Tuesday to push for January 26 celebrations to be moved.

"It's obvious I think, it's time for a national conversation on the date," Mr Gutwein told ABC Radio on Wednesday.

"The fact of the rally in Hobart and also around the country, I think there should be a national conversation."

Mr Gutwein's Liberal colleague, Prime Minister Scott Morrison, believes Australia Day should not be shifted as it marks the date the nation changed forever.

"There is no escaping or cancelling that fact, for better or worse," he said during a keynote address at a flag-raising and citizenship ceremony in Canberra on Tuesday.

Several thousand people went to an Invasion Day rally on the lawns of Hobart's parliament house, a figure organisers have hailed as a record.

Despite Mr Gutwein's comments, Tasmanian Aboriginal Centre campaign coordinator Nala Mansell says the premier is gutless and weak.

She pointed out state Labor and Greens leaders spoke at the rally in favour of changing the date but no Liberal members addressed the crowd.

''The opposition leaders spoke out during Invasion Day rallies about the urgent need to change the date of the Australia Day celebrations," Ms Mansell said.

"Local councils have been changing the dates of their celebrations across Tasmania for years."

January 26 marks the raising of the Union Jack for the first time in 1788 after the First Fleet arrived in Botany Bay the week prior.

The 2021 Australian of the Year, Grace Tame, has voiced her support for moving Australia Day, but said Indigenous voices must be listened to on the issue.

Ms Tame, the first Tasmanian to win the award, has campaigned to give a voice to survivors of sexual abuse after she was raped as a teenager by her school teacher.

Mr Gutwein described her acknowledgement as a "watershed moment".

Australian Associated Press